2.08 Sum and product of roots
x = 6 and x = 9 are the roots of a quadratic equation.
Find the equation in factored form.
Find the equation in expanded form.
x = \dfrac{8}{17} and x = 8 are the roots of a monic quadratic equation.
Find the equation in factored form.
Find the equation in expanded form.
For each of the following equations:
Find the sum of the roots of the equation.
Find the product of the roots of the equation.
If \alpha and \beta are the roots of the equation x^{2} - 8 x + 12 = 0, find the values of:
\alpha + \beta
\alpha \times \beta
\alpha^{2} + \beta^{2}
\dfrac{1}{\alpha} + \dfrac{1}{\beta}
If \alpha and \beta are the roots of the equation 2 x^{2} + 10 x + 8 = 0, find the values of:
\alpha + \beta
\alpha \times \beta
\alpha^{2} + \beta^{2}
\dfrac{1}{\alpha} + \dfrac{1}{\beta}
\alpha^{2} \beta + \alpha \beta^{2}
\left(\alpha - 10\right) \left(\beta - 10\right)
\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}
Consider the equation x^{2} + m x + 18 = 0
Find the sum of the roots in terms of m.
Find the product of the roots.
Find the value of m when the product of the roots is equal to 6 times the sum.
The equation 8 x^{2} + 144 x + m = 0 has two roots, with one being greater than the other by 4. Let the roots be \alpha and \alpha + 4.
Find the sum of the roots.
Find the product of the roots in terms of m.
Find the roots of the equation.
Find the value of m.
Consider the equation 7 x^{2} + 42 x + m = 0.
Find the sum of the roots.
Find the product of the roots in terms of m.
Find the value of m if the roots are reciprocals of each other.
Consider the equation m x^{2} - \left(25 + m\right) x + 225 = 0.
Find the sum of the roots in terms of m.
Find the product of the roots in terms of m.
Find the value of m when the roots are equal in magnitude but opposite in sign.
Hence, find the roots of the equation.
The equation mx^{2} + 9 x + 8 m = 0 has two roots, with one being twice the value of the other. Let the roots be \alpha and 2 \alpha.
Find \alpha.
Find the values of m.
Given the equation k x^{2} + \left(k - 8\right) x - 2 = 0, find the value of k if the sum of the roots is 3.
If \dfrac{4}{3} is one of the roots of 2 x^{2} + m x + 8 = 0, solve for b, the other root of the equation.
Consider the equation x^{2} + n x + 15 = 0, with the smaller root being \alpha. If one root is 2 more than the other, find the possible values of n.
The roots of the equation x^{2} + m x + 9 = 0 are in the ratio 4:1. Find the possible values of m.
Consider the quadratic equation 3 x^{2} + 5 x + 4 = 0, with roots \alpha and \beta.
Determine \alpha^{2} + \beta^{2}.
Determine \alpha^{2} \beta^{2}.
Hence form a quadratic equation whose roots are \alpha^{2} and \beta^{2}.
Consider the quadratic equation 3 x^{2} - 4 x - 2 = 0, with roots \alpha and \beta. Form a quadratic equation whose roots are \dfrac{1}{\alpha} and \dfrac{1}{\beta}.
Consider the quadratic equation x^{2} + 2 x - 5 = 0, with roots \alpha and \beta. Form a quadratic equation whose roots are \alpha + 3 and \beta + 3.
\alpha and \beta are the roots of the equation x^{2} + m x + n = 0.
Express \alpha + \beta in terms of m.
Express \alpha \beta in terms of n.
For the equation n x^{2} + \left( 2 n - m^{2}\right) x + n = 0, write an expression for the sum of the roots in terms of \alpha and \beta.
For the equation n x^{2} + \left( 2 n - m^{2}\right) x + n = 0, write an expression for the product of the roots.
Hence state the roots of the equation n x^{2} + \left( 2 n - m^{2}\right) x + n = 0 in terms of \alpha and \beta.