CAIE A-Level Mathematics 1.1.3 Quadratic Equations and Inequalities

Practise choosing factorisation, completing the square or the quadratic formula to obtain exact roots and using sign analysis to solve quadratic inequalities.

Exam points

choose factorisation, completing the square or the formula and retain exact surd roots
solve a quadratic inequality by locating critical roots and testing the sign between them
check domains and reject roots introduced when rearranging logarithmic or rational equations

[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.

It is given instead that a=5 and that the roots of the equation f(x)=0 are k and -2 k, where k is a constant.

Find the values of b and k.

$(x-k)(x+2k)\equiv x^2+5x+b.$
M1

Comparing coefficients of x:
k=5.
A1

$b=-2k^2=-50.$
A1

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1.1.3—Quadratic equations question 1