Question

8. f(x) = x² + 3x - 2.

Given that a and Bare the roots of the equation f(x) = 0,
find a quadratic equation, with integer coefficients, with roots.
(a) a^2 and ß^2.
(b) (a - ß)^2 and (a + ß)^2.​

Answer 1

(a) x² − 13x + 4

(b) x² − 26x + 153

Explanation:

f(x) = x² + 3x − 2

a + ß = -3

aß = -2

(a) (x − a²) (x − ß²)

x² − (a² + ß²) x + a²ß²

x² − ((a + ß)² − 2aß) x + a²ß²

x² − ((-3)² − 2(-2)) x + (-2)²

x² − (9 + 4) x + 4

x² − 13x + 4

(b) (x − (a − ß)²) (x − (a + ß)²)

x² − ((a − ß)² + (a + ß)²) x + (a − ß)²(a + ß)²

x² − (a² − 2aß + ß² + a² + 2aß + ß²) x + (a² − 2aß + ß²)(a + ß)²

x² − (2a² + 2ß²) x + (a² − 2aß + ß²)(a + ß)²

x² − 2(a² + ß²) x + (a² − 2aß + ß²)(a + ß)²

x² − 2((a + ß)² − 2aß) x + ((a + ß)² − 4aß)(a + ß)²

x² − 2((-3)² − 2(-2)) x + ((-3)² − 4(-2))(-3)²

x² − 2(9 + 4) x + (9 + 8)(9)

x² − 26x + 153