Quadratic Identities

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Question 1 of 6

Find the values of

A

$A$ ,

B

$B$ and

C

$C$ , given that:

x^{2} + 6 x - 2 \equiv A {(x + 1)}^{2} + B (x + 1) + C

$x^2+6x-2≡A(x+1)^2+B(x+1)+C$

$A =$ $A=$ (1)

$B =$ $B=$ (4)

$C =$ $C=$ (-7)

Question 2 of 6

Find the values of

A

$A$ ,

B

$B$ and

C

$C$ , given that:

2 x^{2} + 5 x - 2 \equiv A {(x + 2)}^{2} + B (x + 2) + C

$2x^2+5x-2≡A(x+2)^2+B(x+2)+C$

$A =$ $A=$ (2)

$B =$ $B=$ (-3)

$C =$ $C=$ (-4)

Question 3 of 6

Find the values of

A

$A$ ,

B

$B$ and

C

$C$ , given that:

- x^{2} + 5 x - 2 \equiv A {(x + 3)}^{2} + B (x + 3) + C

$-x^2+5x-2≡A(x+3)^2+B(x+3)+C$

$A =$ $A=$ (-1)

$B =$ $B=$ (11)

$C =$ $C=$ (-26)

Question 4 of 6

Find the values of

P

$P$ ,

Q

$Q$ and

R

$R$ , given that:

x^{2} - 3 \equiv P {(x - 3)}^{2} + Q (x + 1) - 2 R

$x^2-3≡P(x-3)^2+Q(x+1)-2R$

$P =$ $P=$ (1)

$Q =$ $Q=$ (6)

$R =$ $R=$ (9)

Question 5 of 6

Find the values of

$P$ ,

$Q$ and

$R$ , given that:

$2x^2+5x-3≡Px(x-5)+(Qx-1)(R+1)$

$P=$ (2)

$Q=$ (5)

$R=$ (2)

Question 6 of 6

6. Question
Find the values of $K$ , $L$ and $M$ , given that:

$9x^2-6x+2≡(Kx-3)^2+L(x-3)+M$
- $K=$ (3, -3) and (3, -3)
  
  $L=$ (12, -24) and (12, -24)
  
  $M=$ (29, -79) and (29, -79)
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Standard Form

$ax^2+bx+c$

A Quadratic Identity is composed of two quadratic equations that exactly match each other.

First, list the coefficients of the quadratic equation on the left side

$9x^2-6x+2$

$a=9$ $b=-6$ $c=2$

Next, expand the right side of the identity and arrange it in standard form

$(Kx-3)^2+L(x-3)+M$

$=$ $K^2x^2-6Kx+9+Lx-3L+M$

$=$ $K^2x^2-6Kx+Lx+9-3L+M$

$=$ $K^2$ $x^2+($ $-6K+L$ $)x+$ $9-3L+M$

Equate each corresponding coefficient to solve for $A$ , $B$ and $C$

Coefficient of $x^2$ :

$K^2$ $x^2+(-6K+L)x+9-3L+M$

$9$ $x^2-6x+2$

$K^2$ $=$ $9$

$sqrt(K^2)$ $=$ $sqrt9$ Take the square root of both sides

$K$ $=$ $+-3$

$K$ $=$ $3$

$K$ $=$ $-3$

Coefficient of $x$ :

$K^2x^2+($ $-6K+L$ $)x+9-3L+M$

$9x^2$ $-6$ $x+2$

Substitute $K=3$

$-6K+L$ $=$ $-6$

$-6(3)+L$ $=$ $-6$

$-18+L$ $=$ $-6$

$-18+L$ $+18$ $=$ $-6$ $+18$

$L$ $=$ $12$

Substitute $K=-3$

$-6K+L$ $=$ $-6$

$-6(-3)+L$ $=$ $-6$

$18+L$ $=$ $-6$

$18+L$ $-18$ $=$ $-6$ $-18$

$L$ $=$ $-24$

Constants:

$K^2x^2+(-6K+L)x+$ $9-3L+M$

$9x^2-6x+$ $2$

Substitute $L=12$

$9-3L+M$ $=$ $2$

$9-3(12)+M$ $=$ $2$

$9-36+M$ $=$ $2$

$-27+M$ $=$ $2$

$-27+M$ $+27$ $=$ $2$ $+27$

$M$ $=$ $29$

Substitute $L=-24$

$9-3L+M$ $=$ $2$

$9-3(-24)+M$ $=$ $2$

$9+72+M$ $=$ $2$

$81+M$ $=$ $2$

$81+M$ $-81$ $=$ $2$ $-81$

$M$ $=$ $-79$

$K=3$ and $K=-3$

$L=12$ and $L=-24$

$M=29$ and $M=-79$

		$A {(x + 1)}^{2} + B (x + 1) + C$ $A(x+1)^2+B(x+1)+C$
	$=$ $=$	$A (x^{2} + 2 x + 1) + B (x + 1) + C$ $A(x^2+2x+1)+B(x+1)+C$
	$=$ $=$	$A x^{2} + 2 A x + A + B x + B + C$ $Ax^2+2Ax+A+Bx+B+C$
	$=$ $=$	$A x^{2} + 2 A x + B x + A + B + C$ $Ax^2+2Ax+Bx+A+B+C$
	$=$ $=$	$A$ $A$ $x^{2} + ($ $x^2+($ $2 A + B$ $2A+B$ $) x +$ $)x+$ $A + B + C$ $A+B+C$

$2 A + B$ $2A+B$	$=$ $=$	$6$ $6$
$2 (1) + B$ $2(1)+B$	$=$ $=$	$6$ $6$	Substitute $A$ $A$
$2 + B$ $2+B$	$=$ $=$	$6$ $6$
$2 + B$ $2+B$ $- 2$ $-2$	$=$ $=$	$6$ $6$ $- 2$ $-2$	Subtract $2$ $2$ from both sides
$B$ $B$	$=$ $=$	$4$ $4$

$A + B + C$ $A+B+C$	$=$ $=$	$- 2$ $-2$
$1 + 4 + C$ $1+4+C$	$=$ $=$	$- 2$ $-2$	Substitute $A$ $A$ and $B$ $B$
$5 + C$ $5+C$	$=$ $=$	$- 2$ $-2$
$5 + C$ $5+C$ $- 5$ $-5$	$=$ $=$	$- 2$ $-2$ $- 5$ $-5$	Subtract $5$ $5$ from both sides
$C$ $C$	$=$ $=$	$- 7$ $-7$

		$A {(x + 2)}^{2} + B (x + 2) + C$ $A(x+2)^2+B(x+2)+C$
	$=$ $=$	$A (x^{2} + 4 x + 4) + B (x + 2) + C$ $A(x^2+4x+4)+B(x+2)+C$
	$=$ $=$	$A x^{2} + 4 A x + 4 A + B x + 2 B + C$ $Ax^2+4Ax+4A+Bx+2B+C$
	$=$ $=$	$A x^{2} + 4 A x + B x + 4 A + 2 B + C$ $Ax^2+4Ax+Bx+4A+2B+C$
	$=$ $=$	$A$ $A$ $x^{2} + ($ $x^2+($ $4 A + B$ $4A+B$ $) x +$ $)x+$ $4 A + 2 B + C$ $4A+2B+C$

$4 A + B$ $4A+B$	$=$ $=$	$5$ $5$
$4 (2) + B$ $4(2)+B$	$=$ $=$	$5$ $5$	Substitute $A$ $A$
$8 + B$ $8+B$	$=$ $=$	$5$ $5$
$8 + B$ $8+B$ $- 8$ $-8$	$=$ $=$	$5$ $5$ $- 8$ $-8$	Subtract $8$ $8$ from both sides
$B$ $B$	$=$ $=$	$- 3$ $-3$

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Quizzes

$4 A + 2 B + C$ $4A+2B+C$	$=$ $=$	$- 2$ $-2$
$4 (2) + 2 (- 3) + C$ $4(2)+2(-3)+C$	$=$ $=$	$- 2$ $-2$	Substitute $A$ $A$ and $B$ $B$
$8 - 6 + C$ $8-6+C$	$=$ $=$	$- 2$ $-2$
$2 + C$ $2+C$ $- 2$ $-2$	$=$ $=$	$- 2$ $-2$ $- 2$ $-2$	Subtract $2$ $2$ from both sides
$C$ $C$	$=$ $=$	$- 4$ $-4$

		$A {(x + 3)}^{2} + B (x + 3) + C$ $A(x+3)^2+B(x+3)+C$
	$=$ $=$	$A (x^{2} + 6 x + 9) + B (x + 3) + C$ $A(x^2+6x+9)+B(x+3)+C$
	$=$ $=$	$A x^{2} + 6 A x + 9 A + B x + 3 B + C$ $Ax^2+6Ax+9A+Bx+3B+C$
	$=$ $=$	$A x^{2} + 6 A x + B x + 9 A + 3 B + C$ $Ax^2+6Ax+Bx+9A+3B+C$
	$=$ $=$	$A$ $A$ $x^{2} + ($ $x^2+($ $6 A + B$ $6A+B$ $) x +$ $)x+$ $9 A + 3 B + C$ $9A+3B+C$

$6 A + B$ $6A+B$	$=$ $=$	$5$ $5$
$6 (- 1) + B$ $6(-1)+B$	$=$ $=$	$5$ $5$	Substitute $A$ $A$
$- 6 + B$ $-6+B$	$=$ $=$	$5$ $5$
$- 6 + B$ $-6+B$ $+ 6$ $+6$	$=$ $=$	$5$ $5$ $+ 6$ $+6$	Add $6$ $6$ to both sides
$B$ $B$	$=$ $=$	$11$ $11$

$9 A + 3 B + C$ $9A+3B+C$	$=$ $=$	$- 2$ $-2$
$9 (- 1) + 3 (11) + C$ $9(-1)+3(11)+C$	$=$ $=$	$- 2$ $-2$	Substitute $A$ $A$ and $B$ $B$
$- 9 + 33 + C$ $-9+33+C$	$=$ $=$	$- 2$ $-2$
$24 + C$ $24+C$ $- 24$ $-24$	$=$ $=$	$- 2$ $-2$ $- 24$ $-24$	Subtract $24$ $24$ from both sides
$C$ $C$	$=$ $=$	$- 26$ $-26$

		$P {(x - 3)}^{2} + Q (x + 1) - 2 R$ $P(x-3)^2+Q(x+1)-2R$
	$=$ $=$	$P (x^{2} - 6 x + 9) + Q (x + 1) - 2 R$ $P(x^2-6x+9)+Q(x+1)-2R$
	$=$ $=$	$P x^{2} - 6 P x + 9 P + Q x + Q - 2 R$ $Px^2-6Px+9P+Qx+Q-2R$
	$=$ $=$	$P x^{2} - 6 P x + Q x + 9 P + Q - 2 R$ $Px^2-6Px+Qx+9P+Q-2R$
	$=$ $=$	$P$ $P$ $x^{2} + ($ $x^2+($ $- 6 P + Q$ $-6P+Q$ $) x +$ $)x+$ $9 P + Q - 2 R$ $9P+Q-2R$

$-6P+Q$	$=$	$0$
$-6(1)+Q$	$=$	$0$	Substitute $P$
$-6+Q$	$=$	$0$
$-6+Q$ $+6$	$=$	$0$ $+6$	Add $6$ to both sides
$Q$	$=$	$6$

$9P+Q-2R$	$=$	$-3$
$9(1)+6-2R$	$=$	$-3$	Substitute $P$ and $Q$
$15-2R$	$=$	$-3$
$15-2R$ $-15$	$=$	$-3$ $-15$	Subtract $15$ from both sides
$-2R$ $divide(-2)$	$=$	$-18$ $divide(-2)$	Divide both sides by $-2$
$R$	$=$	$9$

		$Px(x-5)+(Qx-1)(R+1)$
	$=$	$Px^2-5Px+QxR+Qx-R-1$
	$=$	$P$ $x^2+($ $-5P+QR+Q$ $)x$ $-R-1$

$-R-1$	$=$	$-3$
$-R-1$ $+1$	$=$	$-3$ $+1$	Add $1$ to both sides
$-R$ $divide(-1)$	$=$	$-2$ $divide(-1)$	Divide both sides by $-1$
$R$	$=$	$2$

$-5P+QR+Q$	$=$	$5$
$-5P+Q(R+1)$	$=$	$5$
$-5(2)+Q(2+1)$	$=$	$5$	Substitute $P$ and $R$
$-10+3Q$	$=$	$5$
$-10+3Q$ $+10$	$=$	$5$ $+10$	Add $10$ to both sides
$3Q$ $divide3$	$=$	$15$ $divide3$	Divide both sides by $3$
$Q$	$=$	$5$

		$(Kx-3)^2+L(x-3)+M$
	$=$	$K^2x^2-6Kx+9+Lx-3L+M$
	$=$	$K^2x^2-6Kx+Lx+9-3L+M$
	$=$	$K^2$ $x^2+($ $-6K+L$ $)x+$ $9-3L+M$

$K^2$	$=$	$9$
$sqrt(K^2)$	$=$	$sqrt9$	Take the square root of both sides
$K$	$=$	$+-3$

$-6K+L$	$=$	$-6$
$-6(3)+L$	$=$	$-6$
$-18+L$	$=$	$-6$
$-18+L$ $+18$	$=$	$-6$ $+18$
$L$	$=$	$12$

$-6K+L$	$=$	$-6$
$-6(-3)+L$	$=$	$-6$
$18+L$	$=$	$-6$
$18+L$ $-18$	$=$	$-6$ $-18$
$L$	$=$	$-24$

$9-3L+M$	$=$	$2$
$9-3(12)+M$	$=$	$2$
$9-36+M$	$=$	$2$
$-27+M$	$=$	$2$
$-27+M$ $+27$	$=$	$2$ $+27$
$M$	$=$	$29$

$9-3L+M$	$=$	$2$
$9-3(-24)+M$	$=$	$2$
$9+72+M$	$=$	$2$
$81+M$	$=$	$2$
$81+M$ $-81$	$=$	$2$ $-81$
$M$	$=$	$-79$