How to solve this ? (Quadratic equations)

Question

It is given that -3 and m-2 are the roots of the quadratic equation X² - (n+1)x + 15 = 0, where m and n are constants. Find the values of m and n.

Answer 1

Since we have roots of -3 and m-2 we can use that to find equivalent quadratic:

(x + 3)(x - m + 2) = 0

x² - mx + 2x + 3x - 3m + 6 = 0

x² + (5-m)x - 3m + 6 = 0

 

Since this is the same as x² - (n+1)x + 15 = 0 we can set coefficients equal:

 

5-m = -n - 1

m-n = 6

 

-3m+6 = 15

-3m = 9

m = -3

 

-3 - n = 6

n = -9

 

check:

x² - (n+1)x + 15 = 0  should have roots -3 and m-2 or -3 and -5

 

x² - (-9+1)x + 15 = 0

x² + 8x + 15 = 0

(x+3)(x+5) = 0

x = -3 or x = -5