Condition on k, prove inequality

A question is this type if and only if it asks to prove (not just find) that a constant satisfies a particular inequality as a consequence of a root condition, requiring algebraic manipulation to derive the inequality from scratch.

3 questions · Standard +0.5

1.02d Quadratic functions: graphs and discriminant conditions

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CAIE P1 2015 November Q8

9 marks Standard +0.3

8 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.

In the case where $a = 6$ and $b = - 8$, find the range of f .
In the case where $a = 5$, the roots of the equation $\mathrm { f } ( x ) = 0$ are $k$ and $- 2 k$, where $k$ is a constant. Find the values of $b$ and $k$.
Show that if the equation $\mathrm { f } ( x + a ) = a$ has no real roots, then $a ^ { 2 } < 4 ( b - a )$.

Edexcel AS Paper 1 Specimen Q10

4 marks Standard +0.3

The equation $kx^2 + 4kx + 3 = 0$, where $k$ is a constant, has no real roots. Prove that $$0 \leqslant k < \frac{3}{4}$$ [4]

WJEC Unit 1 Specimen Q9

7 marks Standard +0.8

The quadratic equation $4x^2 - 12x + m = 0$, where $m$ is a positive constant, has two distinct real roots. Show that the quadratic equation $3x^2 + mx + 7 = 0$ has no real roots. [7]