Grade 11 / Second degree: equations 91 exercises (100% corrected)

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4x28x7(x2−5x24x−1(x−2−4x2−8x204(x234x2−16x64(x−2−10−4x2−16x−12(x−24x2−8x12−4(x−24−4x216x−12−4(x24 ChingQuizz : 13 exercises available for Quizz assessment : 1. Handling second-degree polynomials E.432 Consider the function f whose image of a number x is defined by: f ( x ) = 2 x 2 − 8 x + 6 1 a Establish equality: f ( x )=2 · ( x − 3)( x − 1) b Solve the equation : f ( x )=0 . c Solve the inequation : f ( x ) 0 . 2 a Establish equality: f ( x )+2=2 x − 2 2 b Deduce that, for any real number x , we have : f ( x ) − 2 E.439 Consider the function f whose image of a number x is defined by: f ( x ) = − 4 x 2 + 36 x + 63 1 a Establish equality: f ( x )=(21 − 2 · x )(2 · x +3) b Determine the set of solutions to the equation : f ( x )=0 . c Solve the inequation : f ( x ) > 0 2 a Establish equality: f ( x ) − 144= − 4 · x − 9 2 2 . b Deduce that, for any real number x , we have : f ( x ) 144 E.418 Consider the function f defined on R by: f ( x ) = − 2 · x 2 + 12 · x − 16 1 a Determine the values of the reals a and b realizing the identity: f ( x ) = x − 2 a · x + b b Solve the inequation : f ( x ) < 0 2 a Determine the values of the reals c and d realizing the identity f ( x ) = − 2 · x + c 2 + d b Deduce that, for any real number x , we have : f ( x ) 2 2. Canonical form E.4458 Proposition-Definition: any second-degree polynomial a · x 2 + b · x + c admits an expression of the form : f : x ↦→ ¸ · x − ˛ 2 + ‚ où ¸ , ˛ , ‚ are real numbers with ¸ =0 . This expression is called the vertex form. Associate each second-degree polynomial with its vertex form E.9397 Determine the vertex formof each of the expressions below : a x 2 − 4 x + 1 b x 2 + 6 x + 3 E.9398 Determine the vertex formof each of the following second-degree polynomials: a x 2 + 4 x − 5 b x 2 − 2 x − 1 E.9405 Determine the vertex formof each of the following second-degree polynomials: a x 2 + 2 x − 3 b x 2 − 6 x − 2 c x 2 + 12 x + 5 d x 2 − 10 x + 5 E.4456 Determine the vertex formof each of the following second-degree polynomials: a x 2 + 4 x b x 2 − 14 x + 9 E.4406 Determine the vertex formof each of the following second-degree polynomials: a 2 x 2 + 12 x − 4 b 7 x 2 − 14 x + 10 E.2259 Give the vertex formof each of the second-degree trinomials below : a 2 x 2 + 8 x − 6 b 3 x 2 + 6 x + 6 E.9404 Give the vertex formof each of the second-degree trinomials below : a 2 x 2 + 12 x − 4 b 3 x 2 + 30 x + 12 E.10553 Give the vertex formof each of the second-degree trinomials below : a 9 x 2 + 18 x + 27 b 5 x 2 + 10 x + 2 https://chingmath.fr chapExoCorrec/432 sacados/432 aa chapExoCorrec/439 sacados/439 chapExoCorrec/418 sacados/418 chapExoCorrec/4458 sacados/4458 4x28x7(x2−5x24x−1(x−2−4x2−8x204(x234x2−16x64(x−2−10−4x2−16x−12(x−24x2−8x12−4(x−24−4x216x−12−4(x24 chapExoCorrec/9397 sacados/9397 chapExoCorrec/9398 sacados/9398 chapExoCorrec/9405 sacados/9405 chapExoCorrec/4456 sacados/4456 chapExoCorrec/4406 sacados/4406 Utilisation des quotients dans l'exercice chapExoCorrec/2259 sacados/2259 chapExoCorrec/9404 sacados/9404 chapExoCorrec/10553 sacados/10553
E.2248 Determine the vertex formof each of the expressions below : a x 2 + x + 2 b x 2 − 3 x − 1 E.10552 Determine the vertex formof each of the expressions below : a x 2 + 1 2 x − 3 b x 2 + x − 1 3 E.9396 Determine the vertex formof each of the following second-degree polynomials: a x 2 + 1 4 x + 1 b x 2 + x + 1 E.2250 Let a , b , c be three real numbers. Expand the following expression : a · x + b 2 a 2 − b 2 − 4 ac 4 a E.7101 Determine the vertex formof the polynomial below : 2 · x 2 − 3 · x + 1 3. Canonical form and equations E.7227 Reminder : to solve an equation in the form of the equal-ity of two squares, we use the third remarkable identity to reduce to a product equation : Let’s solve the equation x + 1 2 = 9 : x + 1 2 = 9 x + 1 2 = 3 2 x + 1 2 − 3 2 = 0 D from the remarkable identity: x + 1 + 3 x + 1 − 3 = 0 x + 4 x − 2 = 0 A product is zero if, and only if, at least one of its factors is zero: x + 4 = 0 x = − 4 x − 2 = 0 x = 2 The solution to this equation is : − 4 and 2 . Consider the polynomial ( P ) : x 2 +6 x − 7 1 Determine the vertex formof the polynomial P . 2 Using the vertex formof the polynomial, determine the two solutions of the equation : x 2 +6 x − 7=0 E.9763 Consider the polynomial P =3 · x 2 − 12 · x +17 . 1 Which of the expressions below is the vertex formof the polynomial P : 3 · x − 2 2 +5 3 · x +1 2 +7 3 · x − 3 2 − 17 2 Using the vertex formof the polynomial P , solve the equa-tion : 3 · x 2 − 12 · x +17=8 E.7220 Consider the polynomial ( P ) : x 2 + 4 · x +9 . 1 Determine the values of the numbers a and b realizing the identity: x 2 +4 · x +9= x + a 2 + b 2 Deduce that the equation x 2 +4 · x +9=1 admits no solu-tion. E.10200 1 Give the vertex formof the expression : 3 · x 2 + 12 · x + 2 2 a Solve the equation : 3 · x + 2 2 = 27 b Deduce the solutions of the equation : 3 · x 2 + 12 · x + 2 = 17 E.9442 Consider the equation : ( E ): 2 x 2 + 4 x +4=20 . 1 Determine the vertex formof the polynomial: 2 x 2 +4 x − 16 . 2 Deduce the solutions of the equation ( E ) . E.4410 Consider the expression : ( E ): x 2 + 3 x +10 1 Determine the expression of the vertex formof ( E ) . 2 Deduce that the equation x 2 +3 x +10=0 admits no solu-tion. E.10564 1 Determine the vertex formof the expression : x 2 − 6 x + 3 2 Solve equation : x 2 − 6 x + 3 = 19 E.2246 1 Factor each of the following expressions into a product of factors of the first degree : a 4 x 2 − 81 b x 2 − 5 c (2 x − 4) 2 − 9 d x 2 − 6 · x + 9 2 Find an argument to justify that the expression x 2 +1 cannot be factorized as a product of factors of the first degree. we will then have established the following assertion : There are no real numbers ¸ , ˛ , ‚ , ‹ such that : x 2 + 1 = ( ¸ · x + ˛ )( ‚ · x + ‹ ) E.2247 Consider the expression P defined by: P = x − − 3 + 5 2 x − − 3 − 5 2 1 Give the expanded and reduced form of the expression P . 2 Solve the equation : x 2 +3 · x +1=0 . E.9533 Let be the function f whose image of x is defined by: f ( x ) = x 2 − 2 x − 2 1 Determine the values of the two reals ¸ and ˛ verifying the following equality: f ( x ) = x − ¸ 2 + ˛ 2 Determine the factorized form of the function f . 3 Deduce from the previous question, the antecedents of 0 by the function f . https://chingmath.fr chapExoCorrec/2248 sacados/2248 Utilisation des quotients dans l'exercice chapExoCorrec/10552 sacados/10552 chapExoCorrec/9396 sacados/9396 chapExoCorrec/2250 sacados/2250 chapExoCorrec/7101 sacados/7101 chapExoCorrec/7227 sacados/7227 chapExoCorrec/9763 sacados/9763 chapExoCorrec/7220 sacados/7220 chapExoCorrec/10200 sacados/10200 chapExoCorrec/9442 sacados/9442 chapExoCorrec/4410 sacados/4410 chapExoCorrec/10564 sacados/10564 chapExoCorrec/2246 sacados/2246 chapExoCorrec/2247 sacados/2247 chapExoCorrec/9533 sacados/9533
<0Aucune racine01racine−b2·a>02racine−b−2·a;−b2·a 4. Calculating the discriminant E.4459 Definition: the discriminant of a polynomial a · x 2 + b · x + c of the second degree is a number that is calculated using the coefficients of the polynomial: Δ = b 2 − 4 × a × c Complete the table below for each of the second-degree poly-nomials : a b c Δ= b 2 − 4 · a · c 2 x 2 +5 x +1 − x 2 +7 x +3 x 2 − 5 x +4 2 x 2 − 4 x − 1 − x 2 − x − 1 x 2 +7 E.4408 Determine the discrimants of the second-degree polynomials below : a x 2 + 2 x + 4 b 2 x 2 + 4 x + 1 c x 2 − 2 x + 1 E.10201 Determine the discriminant of each of the second-degree polynomials below : a 3 · x 2 + 5 · x + 2 b − x 2 − 3 · x + 2 c 2 · x 2 − 1 3 · x + 1 E.10554 Determine the discrimants of the second-degree polynomials below : a − 2 x 2 + 2 x + 1 b x 2 − x − 1 c 3 x 2 + x − 2 5. Second-degree equation E.2253 Definition: the roots of a polynomial are the values can-celing this polynomial. Proposition: for a polynomial a · x 2 + b · x + c of the second degree, the number of existing roots depends on the dis-criminant : Determine the roots of the polynomials below : a x 2 + 4 x − 5 b x 2 + x + 1 c 2 x 2 − 13 x + 15 d 3 x 2 − 6 x + 3 E.7082 Determine the roots of the polyno-mials below : a 2 x 2 − 3 x − 2 b − 4 x 2 + 12 x − 9 c 3 x 2 − 4 x + 2 E.770 Determine the roots of the polynomi-als below : a 3 x 2 − 5 x + 6 b 3 x 2 − 24 x + 48 c − 2 · x 2 + x + 6 E.2260 Determine the roots of the following polynomials: a x 2 + 2 x − 15 b 3 x 2 − 5 x + 7 E.10551 Determine the roots of the follow-ing polynomials: a 3 x 2 − 24 x + 48 b − 4 · x 2 − x + 3 E.7086 Determine the roots of the following second-degree polynomials: a 4 · x 2 − 8 · x + 3 b 9 x 2 + 12 x + 4 E.9534 Determine the roots of the following second-degree polynomials: a − 2 · x 2 − 5 · x − 3 b 2 · x 2 + 5 · x + 2 E.10555 Determine the roots of the follow-ing second-degree polynomials: a x 2 + x − 2 b 3 · x 2 + 4 · x + 2 E.5710 Determine the roots, in simplified form, of the following polynomials: a 2 x 2 − 3 x − 9 b 5 x 2 − 8 x + 5 c 2 x 2 − 8 x + 8 E.7104 Solve the following equations : a x 2 − 4 · x − 5 = 0 b 3 · x 2 − x − 2 = 0 E.9464 Determine the roots of the polyno-mials below : a 2 3 · x 2 + x − 3 b − 2 · x 2 − 11 3 · x − 1 c 3 · x 2 − 3 · x + 2 3 https://chingmath.fr chapExoCorrec/4459 sacados/4459 chapExoCorrec/4408 sacados/4408 chapExoCorrec/10201 sacados/10201 chapExoCorrec/10554 sacados/10554 chapExoCorrec/2253 sacados/2253 <0Aucune racine01racine−b2·a>02racine−b−2·a;−b2·a chapExoCorrec/7082 sacados/7082 chapExoCorrec/770 sacados/770 chapExoCorrec/2260 sacados/2260 chapExoCorrec/10551 sacados/10551 chapExoCorrec/7086 sacados/7086 chapExoCorrec/9534 sacados/9534 chapExoCorrec/10555 sacados/10555 chapExoCorrec/5710 sacados/5710 chapExoCorrec/7104 sacados/7104 chapExoCorrec/9464 sacados/9464
ABCDEFG6cm3cm ABCDEFGHI2xx4cm4cm E.9400 Solve the following equations : a 2 7 · x 2 − 5 3 · x − 7 3 = 0 b − 1 2 · x 2 − 4 5 · x + 2 5 = 0 E.7047 Which of the following four statements is correct? The equation − x 3 3 + x 2 +3 · x =0 admits on R : a la solution is − 2 b trois separate solutions c aucune solution d une unique solution 6. Second-degree equation E.10546 Determine the roots of second-degree polynomials: a x 2 − 3 x + 1 b 5 x 2 + 5 x + 1 E.10550 Determine the roots of second-degree polynomials: a x 2 − 7 x + 9 b − 2 x 2 − 3 x + 1 E.10547 Determine the roots of polynomi-als : a − 3 x 2 + 6 x + 1 b 4 x 2 − 6 x + 1 Hint: remember to simplify the radical 7. Second-degree equation and algebraic manipulation E.9399 Solve the following equation : x ( x − 2)( x + 1) = ( x − 2)( − 7 − 3 x ) E.10548 Solve the equation : x − 1 x + 1 = 1 − 2 · x E.10549 Solve the equation : 2 x − 1 2 x + 1 = 1 − x 8. Problems E.10202 Consider the rectangle below with dimensions 6 cm and 3 cm . Inside this rectangle, we construct the square ABCD and the rectangle CEFG whose sides are parallel to the rectangle containing them. Let x be the length of segment [ AB ] . Determine whether it is possible for the hatched part of this figure to have an area of 8 cm 2 . E.10328 Consider the figure below composed : square AEFG , of two rectangles ABCD and CIFH . The points B , D , I , H belong to the sides of the square AEFG . Consider the shaded area shown opposite and note its area A : (measurements are in centimeters) Determine the set of values of x realizing the equation : A = 37 4 Any trace of research or initiative will be taken into account in the assessment. https://chingmath.fr chapExoCorrec/9400 sacados/9400 chapExoCorrec/7047 sacados/7047 Extrait Antilles-Guyanes Juin 2014 chapExoCorrec/10546 sacados/10546 chapExoCorrec/10550 sacados/10550 chapExoCorrec/10547 sacados/10547 chapExoCorrec/9399 sacados/9399 chapExoCorrec/10548 sacados/10548 chapExoCorrec/10549 sacados/10549 chapExoCorrec/10202 sacados/10202 ABCDEFG6cm3cm chapExoCorrec/10328 sacados/10328 ABCDEFGHI2xx4cm4cm
ABCDIxcm10cm ABMNPx19m 5cmxx 9m5mx ABCDMNxx ABCDIJKLxcm7cm5cm E.2955 Consider a square ABCD 10 centime-tres on a side ; a point I belongs to the diagonal [ AC ] , it is marked as shown in the figure below by the length x : From this point I , we construct two squares with respective diago-nals [ AI ] and [ IC ] . Determine the value of x for which the sum of the areas of these two squares is 5 = 8 of the area of the square ABCD . E.9463 Indication : writing and any trace of research will be taken into account when assessing this exercise Consider a segment [ AB ] of length 19 m and a point M be-longing to this segment. Note x the length of segment [ AM ] and place on this figure the points N and P such that AMNP is a square. Determine the value of the number x so that the area of the polygon ABNP is equal to 91 m 2 . E.10190 Consider the square below with side 5 cm and hatched an isosceles right triangle and a right trapezoid : Determine the value of x so that the area of the hatched part measures 16 cm 2 . E.5713 Backing onto his house, Jean has a rectangular-shaped garden with dimensions 9 m and 5 m . He wants to build a driveway on three of the sides of this garden with the same width, and he will plant lawn on the rest of the garden. He proposes the diagram below où the hatched area is the lawn space How wide must the driveway be for the whole lawn to have a surface area of 10 m 2 ? E.5972 In this exercise, any trace of even incomplete research, or of even unsuccessful ini-tiative, will be taken into account in the assessment. Consider a rectangle ABCD whose dimensions are given be-low : AB =6 m ; AD =4 m . For a real number x between 0 and 4 , we place the points M and N respectively on the sides [ AB ] and [ BC ] such that : AM = x ; BN = x Determine the possible value(s) of x so that the area of the triangle MBN is equal to 1 6 of the total area of the rectangle ABCD . E.5706 Consider the figure below où ABCD is a rectangle such that : ABCD is a rectangle. The points I , J , K , L are points belonging respectively to the segments [ AB ] , [ BC ] , [ CD ] and [ AD ] verifying: IB = JC = KD = LA Determine, if possible, the position(s) of point K so that the shaded domain has area 25 cm 2 . E.6786 1 Check that : 10 2 +11 2 +12 2 =13 2 +14 2 2 Are there other series of 5 consecutive natural integers such that the sum of the squares of the two largest equals the sum of the squares of the smallest? E.7152 Let m be a real number. Consider the polynomial P defined by: P =( m +1) · x 2 +(2 − m ) · x +1 For what values of m does the polynomial P admit no roots? https://chingmath.fr chapExoCorrec/2955 sacados/2955 ABCDIxcm10cm chapExoCorrec/9463 sacados/9463 ABMNPx19m chapExoCorrec/10190 sacados/10190 5cmxx chapExoCorrec/5713 sacados/5713 9m5mx chapExoCorrec/5972 sacados/5972 ABCDMNxx chapExoCorrec/5706 sacados/5706 ABCDIJKLxcm7cm5cm chapExoCorrec/6786 sacados/6786 Olympiade Amiens 2015 chapExoCorrec/7152 sacados/7152
ABCDEFxy 5m20m 3I23456JOAB(dMNQP IJOA(dMNQP 9. Problems with substitution E.8384 1 Solve the equation : x 2 − 37 2 · x +85=0 2 Consider a rectangle with 37 m for perimeter and 85 m 2 for area. We’ll note L and ‘ , respectively, the length and width of this rectangle. a Express ‘ as a function of L . b Determine the dimensions of this rectangle. E.10565 The figure below is made up of two rectangles and a square : Note P the polygon ABCDEF . 1 a Express the perimeter L of the polygon P as a func-tion of x and y . b Express the area A of the polygon P as a function of x and y . 2 Determine the set of ordered pairs ( x ; y ) achieving the following two conditions : L = 15 ; A = 14 E.8196 A rectangle has perimeter 19 m and area 12 m 2 . Determine the dimensions of this rectangle. E.8309 Iliam wants to build a rectangular chicken coop in his garden. The figure below shows the dimensions of Iliam’s garden and the location of the chicken coop, which will be built against the wall of his house : The chicken coop is represented by the hatched area. He has 17 m of wire mesh to build his fence and wants to use all of it to build his chicken coop with an area of 30 m 2 . Determine the dimensions of the chicken coop that meet his requirements. 10. Problems and representative curves of functions E.9444 In the plane provided with a O ; I ; J orthonormal, consider the line ( d ) passing through the points A (0 ; 6) and B (3 ; 0) Consider the 4 points M , N , P , Q verifying the following properties : we have the coordinates : Q (2 ; 0) ; M ( x ; 0) where x belongs to the interval 0 ; 2 . points N and P belong to the line ( d ) . points M and P have the same abscissa and points Q and P have the same abscissa. 1 a Determine the expression of the linear function f ad-mitting the straight line ( d ) as its representative curve. b Deduce the coordinates of the point P . c Deduce the expression for the coordinates of point N as a function of x . 2 Note A the area of the trapezoid MNPQ : a Give the expression for the area A as a function of x . b Determine the position of point M so that the area A vale 3 . E.8308 In the plane provided with a O ; I ; J orthonormal, consider the line ( d ) passing through the points J and A (6 ; 0) The point Q has abscissa 4 and the abscissa of the point M belongs to the interval 0 ; 4 ; The point N (resp. P) belongs to the line ( d ) and has the same abscissa as the point M (resp. Q) whose abscissa is noted x . 1 Determine the coordinates of point N as a function of x . 2 Determine the position of point M on the x-axis so that the area of trapezoid MNPQ vale 7 4 . https://chingmath.fr chapExoCorrec/8384 sacados/8384 chapExoCorrec/10565 sacados/10565 ABCDEFxy chapExoCorrec/8196 sacados/8196 chapExoCorrec/8309 sacados/8309 5m20m chapExoCorrec/9444 sacados/9444 3I23456JOAB(dMNQP chapExoCorrec/8308 sacados/8308 IJOA(dMNQP
MNP234567I2JOCf PQMIJO(dA DI2JOC A1A2aI2JOC E.10204 In the plane provided with a refer-ence frame O ; I ; J , consider the curve C f of the function f affine admitting as expression : f ( x ) = − 1 3 · x + 2 Let x be a number belonging to the interval 0 ; 6 . Note N the point on the curve C f of abscissa x and construct the rectangle OMNP whose sides are parallel to the axes. Determine the (or the) coordinates of the point N such that the rectangle OMNP has an area of 5 3 E.6691 In the plane provided with a refer-ence frame O ; I ; J , consider the straight line ( d ) passing through the points A (4 ; 0) and J . Consider a point M belonging to the line ( d ) and with ab-scissa x such that x ∈ 0 ; 4 . Determine the position of the point M on the line ( d ) such that the rectangle OPMQ and the triangle MPA have the same area. Any evidence of research and initiative will be taken into account during assessment. E.6754 Consider the function f de-fined by: f ( x ) = 2 − x for any real x in the interval 0 ; 1 . We admit that : f ( x ) > 0 , for any real x in the interval 0 ; 1 . Note C the representative curve of the function f in an or-thonormal reference frame, and D the plane area between the x-axis and the curve C , on the other hand, between the equa-tion lines x =0 and x =1 . The C curve and D domain are shown opposite. The aim of this exercise is to divide the domain D into two domains of equal area, first by a straight line parallel to the ordinate axis (part A ) , then by a straight line parallel to the abscissa axis (part B ) . Part A Let a be a real such that 0 a 1 . We note A 1 the area of the domain between the curve C , the axis Ox , the straight lines with equations x =0 and x = a , then A 2 that of the domain be-tween the curve C , Ox and the equation lines x = a and x = 1 . A 1 and A 2 are expressed in units of area. Determine the value of a so that the areas A 1 and A 2 are equal. Part B Let b be a positive real. In this part, we propose to divide the domain D into two do-mains of equal area by the straight line of equation y = b . We admit that there exists a single positive real b solution. Determine the value of b . https://chingmath.fr chapExoCorrec/10204 sacados/10204 MNP234567I2JOCf chapExoCorrec/6691 sacados/6691 PQMIJO(dA chapExoCorrec/6754 sacados/6754 DI2JOC A1A2aI2JOC
2IJOMNPA xxyyCfMO E.7153 Consider the function f defined on the interval 0 ; 2 by the relation: f ( x ) = 17 · x − 48 12 · x − 48 The curve C f representative of the function f is represented in the O ; I ; J orthonormal frame below : Let x be a number belonging to the interval 0 ; 2 . We denote by N the point of abscissa x of the curve C f . The point A has coordinates (2 ; 0) and the rectangle OMNP is a rectangle whose sides are parallel to the axes. Determine the value(s) of x so that the triangles JNP and AMN have the same area. E.9465 Consider the function f defined on 0 ; 3 by: f ( x ) = − 3 · x 2 + 9 · x In the plane provided with a reference frame, consider the curve C f representative of the function f and the point M belonging to the curve C : We construct the rectangle having points O and M as oppo-site vertices and whose sides are parallel to the axes of the reference frame. Determine the abscissa of point M so that the area of this rectangle has value 12 . Hint: we can factor the expression for the area of the rect-angle as a function of the abscissa x of the point M by an expression of the form : x − 2 a · x 2 + b · x + c where a;b;c ∈ R 11. Roots and radicals E.9402 Express the two roots of the polyno-mial x 2 − 3 x +1 as a + 5 b where a and b are two real numbers. E.9462 Consider the polynomial P defined by: P = − 4 · x 2 +4 · x +1 Determine the two roots of this polynomial in the form : a + b · 2 where a , b are real numbers. E.8310 Determine the roots of the polyno-mial x 2 +8 x +8 . Its roots are expressed as : a + b · 2 where a;b ∈ R Note: simplification : 32=4 2 E.8288 Determine the roots of the polyno-mials below : a 2 x 2 + 3 x + 3 b x 2 − 2 x − 6 c x 2 + 2 x − 1 Hint: these roots, if any, should be expressed as : a + b · c where a ∈ R , b ∈ R , c ∈ R ∗ + E.8289 Determine the roots of the following quadratic polynomials: a − x 2 − 4 x − 2 b − 4 · x 2 − 12 x − 9 c − 2 · x 2 + 2 · x + 1 Hint: Express the roots in the simplest form possible. E.9401 Determine the roots of the following quadratic polynomials: a − 2 · x 2 − x + 1 b 2 · x 2 − x + 3 c 2 · x 2 − 4 x + 1 Hint: Express the roots in the simplest form possible. E.1969 1 Let ( P ) be the polynomial defined by: x 2 − 2 · x − 1 Evaluate the polynomial ( P ) for x =1+ 2 . 2 Establish that the number − 5 − 17 2 is a root of the poly-nomial x 2 +5 x +2 . 3 Demonstrate that the equation − 3 x 2 +6 x − 2=0 admits as solution set : S = 1 − 3 3 ; 1+ 3 3 12. Sum and product of roots E.2297 1 Theoretical study: We admit that for a trinomial ax 2 + bx + c of the second degree whose discrimINant Δ is strictly positive, these two roots are expressed in the form : https://chingmath.fr chapExoCorrec/7153 sacados/7153 2IJOMNPA chapExoCorrec/9465 sacados/9465 xxyyCfMO chapExoCorrec/9402 sacados/9402 chapExoCorrec/9462 sacados/9462 chapExoCorrec/8310 sacados/8310 chapExoCorrec/8288 sacados/8288 chapExoCorrec/8289 sacados/8289 chapExoCorrec/9401 sacados/9401 chapExoCorrec/1969 sacados/1969 chapExoCorrec/2297 sacados/2297
x 1 = − b − √ Δ 2 a ; x 2 = − b + √ Δ 2 a a Show that the sum of the roots is − b a . b Show that the product of the roots is c a 2 Application : Using the properties established in the previous question, answer the following questions : a Consider the polynomial 2 x 2 +4 x − 16 . After checking that 2 is a root of this polynomial, determine the value of the other root. b Determine a second-degree trinomial admitting two roots whose root sum is 3 and root product is − 10 . 13. Other equations E.8106 Consider the function f defined by: f ( x ) = 4 · x 3 − 18 · x 2 + 16 · x − 4 1 Determine the reals a , b , c realizing the identity: f ( x ) = 2 · x − 1 a · x 2 + b · x + c 2 Draw up the sign table for the function f . E.2255 Consider the polynomial function P of degree 3 defined by: P ( x ) = 3 x 3 + x 2 − 8 x + 4 1 Determine the values of a , b , c such that : P ( x ) = ( x + 2) a · x 2 + b · x + c 2 Deduce the set of zeros of the polynomial P . E.2256 Consider the following rational frac-tion : Q ( x ) = 2 x 2 + 2 x − 8 9 x 2 − 3 x + 1 1 Determine the definition set of the function Q . 2 Determine the set of zeros of this function. 14. Developments and problems E.8195 An item initially costs 128 e , is then increased by t % and finally reduced by t % . Its new price is 126 e . Determine the value of the number t associated with each of these evolutions. E.8194 Neo opened an account in 2017 and deposited the same amount, which we will denote as x , on January 1 er , 2017, January 1 er , 2018, and January 1 er , 2019. His account earns interest at a constant rate of 4 % per year. Knowing that his account is credited, on January 5, 2019, with 468 ; 24 e , , determine the amount x deposited by Neo each year into his account. 15. Unclassified financial years E.2245 Consider the function f defined on R whose image of a number x is defined by the algebraic relation: f ( x ) = 4 x 2 + 4 x − 3 1 a Show that for any x ∈ R , we have : f ( x ) = (2 x − 1)(2 x + 3) b Demonstrate that for any x ∈ R , we have : f ( x ) = (2 x + 1) 2 − 4 2 For each of the following questions, use the most suitable form : a Determine the antecedents of 0 by the function f . b Knowing that the square of a number is always posi-tive or zero, establish that the function f is minorized by − 4 . c Determine the sign of the function f on R . d Solve the inequation : f ( x ) 5 . E.7221 Solve the equation below, giving the answers rounded to the nearest hundredth : 3 · x 2 + x − 1=0 E.8213 Consider the function f defined by the second-degree polynomial: f ( x )=4 x 2 − 12 x +8 1 Determine the vertex formof the expression of f . 2 Establish that at x = 3 2 , the function f admits a mini-mum. The characteristic elements of this extremum will be given. a Verify that the number 1 is a zero of the function f . b Deduce the factorized form of the function f . c Draw up the sign table for the function f . https://chingmath.fr chapExoCorrec/8106 sacados/8106 chapExoCorrec/2255 sacados/2255 chapExoCorrec/2256 sacados/2256 chapExoCorrec/8195 sacados/8195 chapExoCorrec/8194 sacados/8194 chapExoCorrec/2245 sacados/2245 chapExoCorrec/7221 sacados/7221 chapExoCorrec/8213 sacados/8213
-1234567I-2246JOCfCg -1234567I-2246JOCf E.9525 Consider the two functions f and g de-fined on R by: f ( x ) = x 2 − 6 · x + 7 ; g ( x ) = 2 · x 2 − 2 · x + 2 In the plane provided with an orthogonal reference frame O ; I ; J , we give the curves C f and C g representative of the functions f and g respectively. 1 Determine the zeros of the function f . 2 Determine the relative position of the curves C f and C g . E.9526 In the plane provided with an orthogonal reference frame O ; I ; J , consider the curve C f representa-tive of the function f defined on R by: f ( x ) = x 2 − 6 · x + 7 Here is the curve representation C f : Determine the zeros of the function f . E.10203 Let m be a real number ( m ∈ R ) . For any number m , consider the function f m defined on R by: f m ( x ) = 4 · x 2 + 5 − m x + m Determine the set of values of m for which the curve of the function f m intercepts the x-axis only once. E.10563 Proposition: (admitted) A polynomial is factorable in the form of a square, if and only if, its discriminant is zero. Let a be a known number, consider the polynomial P defined by: P = x + 10 x + 10 + a + k Determine the value of k as a function of a so that the poly-nomial P can be factored as a square. Proposal: any trace of research, even if incomplete, will be taken into account during assessment. https://chingmath.fr chapExoCorrec/9525 sacados/9525 -1234567I-2246JOCfCg chapExoCorrec/9526 sacados/9526 -1234567I-2246JOCf chapExoCorrec/10203 sacados/10203 chapExoCorrec/10563 sacados/10563