Year 10: The quadratic equation

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

1. Parts of a quadratic

In $a x ² + b x + c = 0$ :

$a$ 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)

Example:

 $x ² - 5 x + 6 = 0 (x - 2) (x - 3) = 0 \to x = 2 or x = 3$

2.2 Completing the square

Example:

 $x ² + 6 x + 2 = 0 x ² + 6 x = - 2 Add (6 / 2) ² = 9 both sides : x ² + 6 x + 9 = 7 (x + 3) ² = 7 x + 3 = \pm \sqrt 7 x = - 3 \pm \sqrt 7$

2.3 Quadratic formula (works every time)

The superstar formula:

 $x = [- b \pm \sqrt (b ² - 4 a c)] / (2 a)$

Example:

$3 x ² - 4 x - 2 = 0$ ( $a = 3$ , $b = - 4$ , $c = - 2$ ):

 $x = [4 \pm \sqrt ((- 4) ² - 4 \cdot 3 \cdot (- 2))] / 6 = [4 \pm \sqrt (16 + 24)] / 6 = [4 \pm \sqrt 40] / 6 = [4 \pm 2 \sqrt 10] / 6 = (2 \pm \sqrt 10) / 3$

Exercise

3. The discriminant tells the story

The expression under the square root, $Δ = b ² - 4 a c$ , is called the discriminant.

$Δ > 0$ → two distinct real roots (parabola cuts the x‑axis twice).
$Δ = 0$ → one repeated real root (touches the axis).
$Δ < 0$ → 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 $(- b / 2 a, - Δ / 4 a)$ .
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 $a \neq 1$ : When completing the square, divide by $a$ first.
Sign slips in $b ² - 4 a c$ : 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!

Practice!

Here are some exercise quizzes to further sharpen your skills in solving the quadratic equation: