Theory question 102
PolynomialsIf the quadratic equation (2x − 1) − p(x2 + 2) = 0 where p is a constant, has real roots
(a) show that 2p2 + p − 1 ≤ 0;
(b) find the value of p.
Show the model answer
Model answer
(a) Rearrange: px2 − 2x + (2p + 1) = 0
Real roots need b2 − 4ac ≥ 0: 4 − 4p(2p + 1) ≥ 0
1 − 2p2 − p ≥ 0, i.e. 2p2 + p − 1 ≤ 0
(b) (2p − 1)(p + 1) ≤ 0, so −1 ≤ p ≤ ½ (p ≠ 0 for the equation to be quadratic); the boundary values are p = −1 and p = ½
Explanation
Use the discriminant condition b² − 4ac ≥ 0 for real roots.