Question

F(x) = x² + 2ax + a² - a

g(x) = 4 - x²

Let variable a be a real number. Find the range of values of a such that the equation f(x) = g(x) has two different solutions.

***The answer should be -2 < a < 4, but I don't know why.

Answer 1

Explanation:

f(x) = x² + 2ax + a² - a

g(x) = 4 - x²

f(x) = g(x)

x² + 2ax + a² - a = 4 - x²

2x²+2ax+a²-a -4 =0

the equation has two different solutions

=> (2a)² - 4(2)(a²-a-4) > 0

4a²-8a²+8a +32 >0

-4a²+8a+32 > 0

a²-2a-8 >0

(a-4)(a+2) < 0

the answer should be

-2 < a < 4