Hints:
General quadratic equation ax² + bx + c = 0 can be solved using quadratic formula:
x = (−b±√∆)/2a where discriminant ∆ = b²−4ac
For your equation x² + 2px + 2 − p = 0 what is the number in front of x²? Is this equal to a, b or c?
What’s in front of x? Is this equal to a, b or c?
The last part 2 - p is equal to to a, b or c?
For x = (−b ± √∆)/2a to give two real solutions to x² + 2px + 2 − p = 0 the ± will have to come into play. For this to be the case would ∆ = 0, < 0 or > 0? Remember ∆ lies inside √ sign. To help you answer correctly consider a simple parabola which you know crosses X axis twice e.g y = x² - 7x + 12. This means the equation x² - 7x + 12 = 0 has two real solutions. We can find these by factorising and solving for x:
x² - 7x + 12 = 0
(x - 4)(x - 3) = 0
x = 3 or 4
Let’s work out ∆ for x² - 7x + 12 where a = 1, b = -7, c = 12.
∆ = b² − 4ac
= (-7)² - 4(1)(12)
= 49 - 48
= 1
So now we know ∆ > 0 for two solutions.
For your equation
1x² +
2px +
2 − p = 0 you should have found your answers to a,b,c are written in red.
Substitute a = 1, b = 2p, c = 2 − p into expression for ∆:
b² − 4ac > 0
(2p)² − 4(1)(2 − p) > 0
4p² + 4p − 8 > 0
4(p² + p − 2) > 0
p² + p − 2 > 0
(p + 2)(p − 1) > 0
Curve rises above p-axis when p < −2 or p > 1.
∴ x² + 2px + 2 − p = 0 has two roots when p < −2 or p > 1.
Consider what happens when p = 3/2.
x² + 2(3/2)x + 2 − 3/2 = 0
x² + 3x + 1/2 = 0
x = {−3 ± √[3² − 4(1)(1/2)]} / 2(1)
= {−3 ± √(9 − 2)} / 2
= (−3 ± √7) / 2
√7 ≈ 2.646
∴ −3 ± √7 < 0
For p = 3/2 there are two negative roots.
Try p = 2, p = 5/2, p = 3, p= 4
What do you find?
