Year 10: The quadratic discriminant

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Imagine you could peek inside any quadratic equation and know—before doing a single square root—whether it has two solutions, one, or none at all. That crystal‑ball number is the discriminant. Understanding it streamlines solving, graphing, and interpreting quadratics in Year 10 and beyond.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

1. Where does the discriminant live?

Start with the standard quadratic form:

$ax\mathrm{²}+bx+c=0(a\ne 0)$

Inside the quadratic formula, the expression under the square root is $b\mathrm{²}–4ac$. That’s the discriminant, usually written as $\Delta$ (Greek capital delta).

2. How to calculate Δ

Simply substitute the coefficients:

$\Delta =b\mathrm{²}–4ac$

Example for $3x\mathrm{²}–4x–2=0$:

$a=3,b=–4,c=–2\Delta =(–4)\mathrm{²}–4·3·(–2)=16+24=40$

3. Reading the discriminant

• Δ > 0 → two distinct real roots. The parabola crosses the x‑axis twice.
• Δ = 0 → one repeated real root. The parabola just touches (is tangent to) the axis.
• Δ < 0 → no real roots (two complex roots). The parabola sits entirely above or below the axis.

4. Quick examples

$x\mathrm{²}–5x+6=0\Delta =(–5)\mathrm{²}–4·1·6=25–24=1>0\to \text{two}\text{real}\text{roots}(x=2,3)$

$2x\mathrm{²}–12x+18=0\Delta =(–12)\mathrm{²}–4·2·18=144–144=0\to \text{one}\text{repeated}\text{root}(x=3)$

$x\mathrm{²}+4x+13=0\Delta =4\mathrm{²}–4·1·13=16–52=–36<0\to \text{no}\text{real}\text{roots}$

Exercise

5. Visual link to the graph

The discriminant predicts how many times a parabola meets the x‑axis:

• Two roots → two intersection points.
• One root → vertex sits on the axis.
• No real roots → parabola floats completely above or below the axis.

6. Not everything can be negative

: In many situations, the quadratic formula describes a variable that cannot be negative, such as time and length. So, when you find two solutions of such a quadratic equation, one of which is negative, then you should exclude the negative solution.

7. Why bother with Δ?

• Time‑saver: Know if factorising is possible before trying.
• Design checks: Engineers verify whether a trajectory will hit the ground (real roots) or not.
• Error spotting: A negative Δ in a “distance” problem flags a setup mistake; distances can’t be complex!

8. Common mistakes

• Sign slips: Square $b$ first—$(–7)\mathrm{²}=49$, not –49.
• Forgetting 4ac: Multiply all three numbers before subtracting.
• Mixing up roots and intercepts: The discriminant tells you about x‑intercepts, not y‑intercepts.

9. Summary

The discriminant is your diagnostic tool for quadratics—quick, reliable, and packed with information. Next time you face $ax\mathrm{²}+bx+c=0$, calculate Δ first. It will steer your solving strategy and sharpen your graph sketches.