If you have sat an A Math prelim, you will have met this phrase: "Find the range of values of $k$ for which...". Nine times out of ten, that question is testing one small expression, $b^2 - 4ac$, and your ability to solve the quadratic inequality that falls out of it.
This is one of the highest return topics in the whole 4049 syllabus. The logic is short, the marks are predictable, and once you drill the three standard question types, you rarely lose a mark again.
What the Discriminant Actually Tells You
For a quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$, the discriminant is
$$D = b^2 - 4ac$$
Read it like this:
- $D > 0$: two distinct real roots (the curve cuts the $x$-axis twice)
- $D = 0$: two equal real roots, also called a repeated root (the curve touches the $x$-axis, so the line is a tangent)
- $D < 0$: no real roots (the curve never meets the $x$-axis)
- $D \geq 0$: real roots, which includes the equal case
That last line is where marks quietly disappear. "Real and distinct" means strictly greater than zero. "Real" on its own means greater than or equal to zero. Underline the keyword in the question before you write anything.
The "always positive" and "always negative" conditions
These two results are worth memorising exactly as stated:
- $ax^2 + bx + c > 0$ for all real $x$ requires $a > 0$ and $D < 0$
- $ax^2 + bx + c < 0$ for all real $x$ requires $a < 0$ and $D < 0$
Both conditions, every time. A curve with $D < 0$ but $a < 0$ sits entirely below the axis, so it is always negative, not always positive. Examiners award a mark for stating the condition on $a$, so write it down even when it is obvious.
Type 1: Find the Range of Values of k
Question: Find the range of values of $k$ for which $x^2 + 2kx + (3k + 4) > 0$ for all real values of $x$.
Here $a = 1 > 0$, so we only need $D < 0$.
$$(2k)^2 - 4(1)(3k+4) < 0$$ $$4k^2 - 12k - 16 < 0$$ $$k^2 - 3k - 4 < 0$$ $$(k - 4)(k + 1) < 0$$
Now solve the quadratic inequality. Sketch $y = (k-4)(k+1)$: an upward parabola cutting the $k$-axis at $-1$ and $4$. It is below the axis between the roots, so
$$-1 < k < 4$$
Rule of thumb for an upward parabola: less than zero means between the roots, greater than zero means outside the roots. Do not memorise it blindly, sketch the parabola in the margin. It takes four seconds and stops you writing $k < -1$ or $k > 4$ by accident.
Type 2: Line and Curve Intersection
Question: Find the values of $m$ for which the line $y = mx + 2$ meets the curve $y = x^2 + 3x + 6$ at two distinct points.
Step one is always the same: eliminate $y$ and form a quadratic in $x$.
$$x^2 + 3x + 6 = mx + 2$$ $$x^2 + (3 - m)x + 4 = 0$$
Two distinct points means $D > 0$:
$$(3 - m)^2 - 16 > 0$$ $$(3 - m - 4)(3 - m + 4) > 0$$ $$(-1 - m)(7 - m) > 0$$
Multiply both sides by $-1$ and flip the inequality, or expand instead: $m^2 - 6m - 7 > 0$, giving $(m - 7)(m + 1) > 0$. Outside the roots:
$$m < -1 \quad \text{or} \quad m > 7$$
If the question had said "is a tangent to the curve", set $D = 0$ and solve for $m$ exactly. If it said "does not meet the curve", set $D < 0$.
Type 3: The Hidden a = 0 Trap
Question: Find the values of $k$ for which $kx^2 + (k + 3)x + 4 = 0$ has real roots.
The coefficient of $x^2$ contains $k$, so you must state $k \neq 0$ for the equation to be quadratic.
$$(k+3)^2 - 16k \geq 0$$ $$k^2 - 10k + 9 \geq 0$$ $$(k - 1)(k - 9) \geq 0$$ $$k \leq 1 \quad \text{or} \quad k \geq 9, \quad k \neq 0$$
That final exclusion is a real mark in the mark scheme. Scan every parameter question for it.
Common Mistakes to Kill Before Paper 1
- Forgetting to flip the inequality sign when dividing by a negative number.
- Writing $-1 > k > 4$, which is meaningless. Two separate regions need "or", one region needs a sandwich like $-1 < k < 4$.
- Using $D > 0$ when the question says "real roots" (should be $\geq$).
- Solving for $x$ instead of the parameter. Once you form the discriminant, $x$ is gone.
- Ignoring the condition on $a$ in "always positive" questions.
A Five-Minute Drill
Take ten past-paper parts that mention $k$, $m$ or $p$. For each one, write only two lines: the quadratic you formed, and whether you need $D > 0$, $D = 0$, $D < 0$ or $D \geq 0$. Do not solve. You are training the decision, which is where the marks are lost, not the algebra.
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