Year 10: The quadratic equation

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Quadratic equations pop up in everything from footy kicks to finance graphs. Any equation that can be written in the form $ax\mathrm{²}+bx+c=0$ (with $a\ne 0$) is quadratic. Mastering their solutions now will make senior algebra, physics, and calculus far smoother.

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

1. Parts of a quadratic

In $ax\mathrm{²}+bx+c=0$:

• $a$ is the leading coefficient (controls the “width” and direction of the parabola).
• $b$ is the linear coefficient.
• $c$ is the constant term (the y‑intercept when graphed).

2. Three ways to solve

2.1 Factorising (when it splits nicely)

$x\mathrm{²}–5x+6=0(x–2)(x–3)=0\to x=2\text{or}x=3$

2.2 Completing the square

$x\mathrm{²}+6x+2=0x\mathrm{²}+6x=–2\text{Add}(6/2)\mathrm{²}=9\text{both}\text{sides}:x\mathrm{²}+6x+9=7(x+3)\mathrm{²}=7x+3=\pm \surd 7x=–3\pm \surd 7$

2.3 Quadratic formula (works every time)

The superstar formula:

$x=[-b\pm \surd (b\mathrm{²}–4ac\left)\right]/\left(2a\right)$

$x=[4\pm \surd ((-4)\mathrm{²}–4·3·(–2)\left)\right]/6=[4\pm \surd (16+24\left)\right]/6=[4\pm \surd 40]/6=[4\pm 2\surd 10]/6=(2\pm \surd 10)/3$

Exercise

3. The discriminant tells the story

The expression under the square root, $\Delta =b\mathrm{²}–4ac$, is called the discriminant.

• $\Delta >0$ → two distinct real roots (parabola cuts the x‑axis twice).
• $\Delta =0$ → one repeated real root (touches the axis).
• $\Delta <0$ → no real roots (parabola floats above or below the axis).

Exercise

4. Graphing: meet the parabola

A quadratic’s graph is a parabola:

• Opens upward if $a>0$, downward if $a<0$.
• The vertex (turning point) sits at $(–b/2a,–\Delta /4a)$.
• Roots found algebraically are the x‑intercepts.

Exercise

5. Check your answers

Always substitute each root back into the original equation. Zero (within rounding) confirms correctness.

6. Common mistakes

• Forgetting $a\ne 1$: When completing the square, divide by $a$ first.
• Sign slips in $b\mathrm{²}–4ac$: Square $b$ before subtracting; keep an eye on negatives.
• Leaving answers unsimplified: Factor out perfect squares under the root, and reduce fractions.
• Mixing ± signs: Remember each ± gives two separate solutions.

7. Why quadratics matter

Quadratic models describe projectile motion, profit curves, satellite dishes, and braking distances. Grasp the equation now and you’ll wield a tool that engineers, economists, and game designers use daily.

So next time a quadratic appears, pick the best method—factorise if it splits, complete the square for neat algebra, or unleash the quadratic formula for anything—and always confirm with the discriminant and a quick check. Parabolas will soon feel like friendly curves instead of mysterious arches!