Quadratic Identities
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Question 1 of 6
1. Question
Find the values of `A`, `B` and `C`, given that:`x^2+6x-2≡A(x+1)^2+B(x+1)+C`-
`A=` (1)`B=` (4)`C=` (-7)
Hint
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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`x^2+6x-2``a=1` `b=6` `c=-2`Next, expand the right side of the identity and arrange it in standard form`A(x+1)^2+B(x+1)+C` `=` `A(x^2+2x+1)+B(x+1)+C` `=` `Ax^2+2Ax+A+Bx+B+C` `=` `Ax^2+2Ax+Bx+A+B+C` `=` `A``x^2+(``2A+B``)x+``A+B+C` Equate each corresponding coefficient to solve for `A`, `B` and `C`Coefficient of `x^2`:`A``x^2+(2A+B)x+A+B+C``1``x^2+6x-2``A` `=` `1` Coefficient of `x`:`Ax^2+(``2A+B``)x+A+B+C``x^2+``6``x-2``2A+B` `=` `6` `2(1)+B` `=` `6` Substitute `A` `2+B` `=` `6` `2+B` `-2` `=` `6` `-2` Subtract `2` from both sides `B` `=` `4` Constants:`Ax^2+(2A+B)x+``A+B+C``x^2+6x``-2``A+B+C` `=` `-2` `1+4+C` `=` `-2` Substitute `A` and `B` `5+C` `=` `-2` `5+C` `-5` `=` `-2` `-5` Subtract `5` from both sides `C` `=` `-7` `A=1``B=4``C=-7` -
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Question 2 of 6
2. Question
Find the values of `A`, `B` and `C`, given that:`2x^2+5x-2≡A(x+2)^2+B(x+2)+C`-
`A=` (2)`B=` (-3)`C=` (-4)
Hint
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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`2x^2+5x-2``a=2` `b=5` `c=-2`Next, expand the right side of the identity and arrange it in standard form`A(x+2)^2+B(x+2)+C` `=` `A(x^2+4x+4)+B(x+2)+C` `=` `Ax^2+4Ax+4A+Bx+2B+C` `=` `Ax^2+4Ax+Bx+4A+2B+C` `=` `A``x^2+(``4A+B``)x+``4A+2B+C` Equate each corresponding coefficient to solve for `A`, `B` and `C`Coefficient of `x^2`:`A``x^2+(4A+B)x+4A+2B+C``2``x^2+5x-2``A` `=` `2` Coefficient of `x`:`Ax^2+(``4A+B``)x+4A+2B+C``2x^2+``5``x-2``4A+B` `=` `5` `4(2)+B` `=` `5` Substitute `A` `8+B` `=` `5` `8+B` `-8` `=` `5` `-8` Subtract `8` from both sides `B` `=` `-3` Constants:`Ax^2+(4A+B)x+``4A+2B+C``x^2+5x``-2``4A+2B+C` `=` `-2` `4(2)+2(-3)+C` `=` `-2` Substitute `A` and `B` `8-6+C` `=` `-2` `2+C` `-2` `=` `-2` `-2` Subtract `2` from both sides `C` `=` `-4` `A=2``B=-3``C=-4` -
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Question 3 of 6
3. Question
Find the values of `A`, `B` and `C`, given that:`-x^2+5x-2≡A(x+3)^2+B(x+3)+C`-
`A=` (-1)`B=` (11)`C=` (-26)
Hint
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Nice Job!
Incorrect
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`-x^2+5x-2``a=-1` `b=5` `c=-2`Next, expand the right side of the identity and arrange it in standard form`A(x+3)^2+B(x+3)+C` `=` `A(x^2+6x+9)+B(x+3)+C` `=` `Ax^2+6Ax+9A+Bx+3B+C` `=` `Ax^2+6Ax+Bx+9A+3B+C` `=` `A``x^2+(``6A+B``)x+``9A+3B+C` Equate each corresponding coefficient to solve for `A`, `B` and `C`Coefficient of `x^2`:`A``x^2+(6A+B)x+9A+3B+C``-1``x^2+5x-2``A` `=` `-1` Coefficient of `x`:`Ax^2+(``6A+B``)x+9A+3B+C``-x^2+``5``x-2``6A+B` `=` `5` `6(-1)+B` `=` `5` Substitute `A` `-6+B` `=` `5` `-6+B` `+6` `=` `5` `+6` Add `6` to both sides `B` `=` `11` Constants:`Ax^2+(4A+B)x+``9A+3B+C``-x^2+5x``-2``9A+3B+C` `=` `-2` `9(-1)+3(11)+C` `=` `-2` Substitute `A` and `B` `-9+33+C` `=` `-2` `24+C` `-24` `=` `-2` `-24` Subtract `24` from both sides `C` `=` `-26` `A=-1``B=11``C=-26` -
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Question 4 of 6
4. Question
Find the values of `P`, `Q` and `R`, given that:`x^2-3≡P(x-3)^2+Q(x+1)-2R`-
`P=` (1)`Q=` (6)`R=` (9)
Hint
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Well Done!
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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`x^2-3``a=1` `b=0` `c=-3`Next, expand the right side of the identity and arrange it in standard form`P(x-3)^2+Q(x+1)-2R` `=` `P(x^2-6x+9)+Q(x+1)-2R` `=` `Px^2-6Px+9P+Qx+Q-2R` `=` `Px^2-6Px+Qx+9P+Q-2R` `=` `P``x^2+(``-6P+Q``)x+``9P+Q-2R` Equate each corresponding coefficient to solve for `P`, `Q` and `R`Coefficient of `x^2`:`P``x^2+(-6P+Q)x+9P+Q-2R``1``x^2-3``P` `=` `1` Coefficient of `x`:`Px^2+(``-6P+Q``)x+9P+Q-2R``x^2-3``-6P+Q` `=` `0` `-6(1)+Q` `=` `0` Substitute `P` `-6+Q` `=` `0` `-6+Q` `+6` `=` `0` `+6` Add `6` to both sides `Q` `=` `6` Constants:`Px^2+(-6P+Q)x+``9P+Q-2R``x^2``-3``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` `P=1``Q=6``R=9` -
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Question 5 of 6
5. Question
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)
Hint
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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`2x^2+5x-3``a=2` `b=5` `c=-3`Next, expand the right side of the identity and arrange it in standard form`Px(x-5)+(Qx-1)(R+1)` `=` `Px^2-5Px+QxR+Qx-R-1` `=` `P``x^2+(``-5P+QR+Q``)x``-R-1` Equate each corresponding coefficient to solve for `P`, `Q` and `R`Coefficient of `x^2`:`P``x^2+(-6P+Q)x+9P+Q-2R``2``x^2+5x-3``P` `=` `2` Constants:`Px^2+(-5P+QR+Q)x``-R-1``2x^2+5x``-3``-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` Coefficient of `x`:`Px^2+(``-5P+QR+Q``)x-R-1``2x^2+``5``x-3``-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` `P=2``Q=5``R=2` -
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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)
Hint
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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` -
Quizzes
- Sum & Product of Roots 1
- Sum & Product of Roots 2
- Sum & Product of Roots 3
- Sum & Product of Roots 4
- Solving Equations by Factoring 1
- Solving Equations Using the Quadratic Formula
- Completing the Square 1
- Completing the Square 2
- Intro to Quadratic Functions (Parabolas) 1
- Intro to Quadratic Functions (Parabolas) 2
- Intro to Quadratic Functions (Parabolas) 3
- Graph Quadratic Functions in Standard Form 1
- Graph Quadratic Functions in Standard Form 2
- Graph Quadratic Functions by Completing the Square
- Graph Quadratic Functions in Vertex Form
- Write a Quadratic Equation from the Graph
- Write a Quadratic Equation Given the Vertex and Another Point
- Quadratic Inequalities 1
- Quadratic Inequalities 2
- Quadratics Word Problems 1
- Quadratics Word Problems 2
- Quadratic Identities
- Graphing Quadratics Using the Discriminant
- Positive and Negative Definite
- Applications of the Discriminant 1
- Applications of the Discriminant 2
- Solving Reducible Equations