CAIE A-Level Mathematics 1.1.2 Quadratic Discriminant

Practise forming b² − 4ac after rearranging to quadratic form and solving parameter inequalities for two distinct, one repeated or no real roots.

Exam points

rearrange to ax² + bx + c = 0 before identifying the correct coefficients
set b² − 4ac above, equal to or below zero for the stated number of real roots
solve the resulting parameter inequality with correct strict endpoints and interval notation

[Maximum number: 3]

The function f is defined, for $x \in \mathbb{R}$ , by $\mathrm{f}: x \mapsto x^{2}+a x+b$ , where a and b are constants.

Show that if the equation f(x+a)=a has no real roots then $a^{2}<4(b-a)$ .

$(x+a)^2+a(x+a)+b=a.$
M1*

Use the discriminant for the resulting quadratic:
$9a^2-4(2a^2+b-a)<0.$
DM1

Hence
$a^2<4(b-a).$
A1 AG

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