Sum & Product of Roots 3
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Question 1 of 5
1. Question
If the roots of `2x^2-3x-2=0` are `alpha` and `beta`, find:`alpha^2+beta^2`Hint
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Sum of Roots `alpha` and `beta`
$$\alpha+\beta=-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$Product of Roots `alpha` and `beta`
$$\alpha\beta=\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$First, list the coefficients of the quadratic equation individually`2x^2-3x-2``a=2` `b=-3` `c=-2`Slot the coefficients to the Sum and Product of Roots FormulaSum of Roots:`alpha+beta` `=` $$-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$ `=` $$-\frac{\color{#9a00c7}{-3}}{\color{#00880A}{2}}$$ `=` `3/2` Product of Roots:`alphabeta` `=` $$\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$ `=` $$\frac{\color{#007DDC}{-2}}{\color{#00880A}{2}}$$ `=` `-1` Manipulate the given expression until you can substitute the sum and product of roots.`alpha^2+beta^2` `=` `alpha^2` `+2alphabeta``+beta^2` `-2alphabeta` `=` `(alpha+beta)^2-2alphabeta` `=` `(3/2)^2-2(-1)` `=` `9/4+2` `=` `17/4` `=` `4 1/4` `alpha^2+beta^2=4 1/4` -
Question 2 of 5
2. Question
If the roots of `2x^2-3x-2=0` are `alpha` and `beta`, find:`alpha^2beta^3+beta^2alpha^3`Write fractions in the format “a/b”- (3/2)
Hint
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Sum of Roots `alpha` and `beta`
$$\alpha+\beta=-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$Product of Roots `alpha` and `beta`
$$\alpha\beta=\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$First, list the coefficients of the quadratic equation individually`2x^2-3x-2``a=2` `b=-3` `c=-2`Slot the coefficients to the Sum and Product of Roots FormulaSum of Roots:`alpha+beta` `=` $$-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$ `=` $$-\frac{\color{#9a00c7}{-3}}{\color{#00880A}{2}}$$ `=` `3/2` Product of Roots:`alphabeta` `=` $$\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$ `=` $$\frac{\color{#007DDC}{-2}}{\color{#00880A}{2}}$$ `=` `-1` Manipulate the given expression until you can substitute the sum and product of roots.`alpha^2beta^3+beta^2alpha^3` `=` `alpha^2beta^2(alpha+beta)` `=` `(alphabeta)^2(alpha+beta)` `=` `(-1)^2(3/2)` `=` `1(3/2)` `=` `3/2` `alpha^2beta^3+beta^2alpha^3=3/2` -
Question 3 of 5
3. Question
If the roots of `x^2-5x+1=0` are `alpha` and `beta`, find:`alpha^2+beta^2`- (23)
Hint
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Nice Job!
Incorrect
Sum of Roots `alpha` and `beta`
$$\alpha+\beta=-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$Product of Roots `alpha` and `beta`
$$\alpha\beta=\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$First, list the coefficients of the quadratic equation individually`x^2-5x+1=0``a=1` `b=-5` `c=1`Slot the coefficients to the Sum and Product of Roots FormulaSum of Roots:`alpha+beta` `=` $$-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$ `=` $$-\frac{\color{#9a00c7}{-5}}{\color{#00880A}{1}}$$ `=` `5` Product of Roots:`alphabeta` `=` $$\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$ `=` $$\frac{\color{#007DDC}{1}}{\color{#00880A}{1}}$$ `=` `1` Manipulate the given expression until you can substitute the sum and product of roots.`alpha^2+beta^2` `=` `(alpha+beta)^2-2alphabeta` `=` `(5)^2-2(1)` `=` `25-2` `=` `23` `alpha^2+beta^2=23` -
Question 4 of 5
4. Question
If the roots of `x^2-5x+1=0` are `alpha` and `beta`, find:`alpha^2beta^3+beta^2alpha^3`- (5)
Hint
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Excellent!
Incorrect
Sum of Roots `alpha` and `beta`
$$\alpha+\beta=-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$Product of Roots `alpha` and `beta`
$$\alpha\beta=\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$First, list the coefficients of the quadratic equation individually`x^2-5x+1=0``a=1` `b=-5` `c=1`Slot the coefficients to the Sum and Product of Roots FormulaSum of Roots:`alpha+beta` `=` $$-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$ `=` $$-\frac{\color{#9a00c7}{-5}}{\color{#00880A}{1}}$$ `=` `5` Product of Roots:`alphabeta` `=` $$\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$ `=` $$\frac{\color{#007DDC}{1}}{\color{#00880A}{1}}$$ `=` `1` Manipulate the given expression until you can substitute the sum and product of roots.`alpha^2beta^3+beta^2alpha^3` `=` `alpha^2beta^2(alpha+beta)` `=` `(alphabeta)^2(alpha+beta)` `=` `(1)^2(5)` `=` `5` `alpha^2beta^3+beta^2alpha^3=5` -
Question 5 of 5
5. Question
If the roots of `2x^2-4x-1=0` are `alpha` and `beta`, find:`alpha^2+beta^2`- (5)
Hint
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Fantastic!
Incorrect
Sum of Roots `alpha` and `beta`
$$\alpha+\beta=-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$Product of Roots `alpha` and `beta`
$$\alpha\beta=\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$First, list the coefficients of the quadratic equation individually`2x^2-4x-1=0``a=2` `b=-4` `c=-1`Slot the coefficients to the Sum and Product of Roots FormulaSum of Roots:`alpha+beta` `=` $$-\frac{\color{#9a00c7}{b}}{\color{#00880A}{a}}$$ `=` $$-\frac{\color{#9a00c7}{-4}}{\color{#00880A}{2}}$$ `=` `2` Product of Roots:`alphabeta` `=` $$\frac{\color{#007DDC}{c}}{\color{#00880A}{a}}$$ `=` $$\frac{\color{#007DDC}{-1}}{\color{#00880A}{2}}$$ `=` `-1/2` Manipulate the given expression until you can substitute the sum and product of roots.`alpha^2+beta^2` `=` `(alpha+beta)^2-2alphabeta` `=` `(2)^2-2(-1/2)` `=` `4+1` `=` `5` `alpha^2+beta^2=5`
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