Sum & Product of Roots 4
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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)`- (0)
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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)` `=` `alphabeta-2alpha-2beta+4` `=` `alphabeta-2(alpha+beta)+4` `=` `(-1)-2(3/2)+4` `=` `-1-3+4` `=` `0` `(alpha-2)(beta-2)=0` -
Question 2 of 5
2. Question
If the roots of `x^2-5x+1=0` are `alpha` and `beta`, find:`4/alpha+4/beta`- (20)
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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`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.`4/alpha+4/beta` `=` `(4alpha+4beta)/(alphabeta)` `=` `(4(alpha+beta))/(alphabeta)` `=` `(4(5))/(1)` `=` `20` `4/alpha+4/beta=20` -
Question 3 of 5
3. Question
If the roots of `x^2-5x+1=0` are `alpha` and `beta`, find:`(alpha-3)(beta-3)`- (-5)
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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`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-3)(beta-3)` `=` `alphabeta-3alpha-3beta+9` `=` `alphabeta-3(alpha+beta)+9` `=` `1-3(5)+9` `=` `1-15+9` `=` `-5` `(alpha-3)(beta-3)=-5` -
Question 4 of 5
4. Question
If the roots of `2x^2-4x-1=0` are `alpha` and `beta`, find:`1/alpha+1/beta`- (-4)
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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^2-4x-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.`1/alpha+1/beta` `=` `(alpha+beta)/(alphabeta)` `=` `2/(-1/2)` `=` `-4` `1/alpha+1/beta=-4` -
Question 5 of 5
5. Question
If the roots of `2x^2-4x-1=0` are `alpha` and `beta`, find:`alpha/beta+beta/alpha`- (-10)
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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^2-4x-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.Remember from the previous question that `alpha^2+beta^2=5``alpha/beta+beta/alpha` `=` `(alpha^2+beta^2)/(alphabeta)` `=` `5/(-1/2)` `=` `-10` `alpha/beta+beta/alpha=-10`
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