Chapter 4 · Introduction

Chapter 4 — Quadratic Equations

When the unknown gets multiplied by itself — an area, a falling ball, a speed and a time — the equation becomes quadratic. Learn to set it up, solve it and tell in advance how many answers to expect.

On Diwali night a rocket shoots up from a bottle on the terrace, slows down, stops for an instant at the top and falls back. Suppose its height after $t$ seconds is $h=30t-5t^2$ metres. When is it $40$ m above the ground? Put $h=40$ and you get $30t-5t^2=40$, or $5t^2-30t+40=0$. The unknown $t$ appears squared, so this is not a linear equation — it is a quadratic equation. And it has two answers, $t=2$ s and $t=4$ s: once on the way up and once on the way down.

Quadratic equations appear whenever a quantity is multiplied by itself or by something that depends on it: the area of a plot whose length depends on its breadth, the time for a journey when the speed changes, the height of a ball, the price and the number of items you can buy with a fixed amount of money. In this chapter you learn to turn such situations into an equation $ax^2+bx+c=0$, to solve it by factorisation, and to solve any quadratic with the quadratic formula. Finally the discriminant $D=b^2-4ac$ tells you, before you solve, whether there are two different answers, one repeated answer or no real answer at all.

Word problem → equation→Standard form $ax^2+bx+c=0$→Factorise: split the middle term→Quadratic formula→Discriminant: how many real roots?
🎯 By the end of this chapter you can…
  • decide whether an equation is quadratic by simplifying it to the standard form $ax^2+bx+c=0$ with $a\ne0$;
  • check whether a given number is a root of a quadratic equation, and find an unknown coefficient from a known root;
  • model word problems about areas, numbers, ages, speeds and costs by quadratic equations, and reject answers that make no sense in the situation;
  • solve quadratic equations by factorisation (splitting the middle term) and check the roots;
  • solve any quadratic equation with the quadratic formula, and use the discriminant $D=b^2-4ac$ to decide whether its roots are real and distinct, real and equal, or not real.
Teaser — When Is the Rocket at That Height?

The rocket's height is $h=30t-5t^2$ metres after $t$ seconds. Choose a target height $H$ (the amber line). Each moment the rocket is exactly at height $H$ is a root of the quadratic equation $5t^2-30t+H=0$; those moments are marked in red. Press Play to launch. Try $H=45$ and $H=50$: how many roots do you get now?

time $t$ (s)—
height $h$ (m)—
equation—
roots found—

Sections in this chapter

4.1

Quadratic Equations and Modelling

What makes an equation quadratic, the standard form, checking whether a number is a root, and turning word problems into equations.

$ax^2+bx+c=0,\ a\ne0$
Open section →4.2

Solving Quadratic Equations by Factorisation

Split the middle term, factorise, and use "a product is zero only when a factor is zero" to find both roots.

$(x-\alpha)(x-\beta)=0\Rightarrow x=\alpha$ or $x=\beta$
Open section →4.3

The Quadratic Formula and the Nature of Roots

One formula that solves every quadratic, and the discriminant that tells you in advance whether there are two, one or no real roots.

$x=\frac{-b\pm\sqrt{D}}{2a},\ D=b^2-4ac$
Open section →
✅ Before you start
You need: quadratic polynomials $ax^2+bx+c$, their zeroes and the picture of a parabola meeting the $x$-axis (Section 2.1); splitting the middle term and the relations $\alpha+\beta=-\tfrac ba$, $\alpha\beta=\tfrac ca$ (Section 2.2); and turning a word problem into equations (Section 3.3). From Class 9 you need the identities $(a\pm b)^2=a^2\pm2ab+b^2$ and $a^2-b^2=(a+b)(a-b)$, the area of a rectangle and the relation speed $=\dfrac{\text{distance}}{\text{time}}$. Warm up: factorise $x^2+2x-15$ (answer: $(x+5)(x-3)$).
← 3.3 The Elimination Method and Word Problems4.1 Quadratic Equations and Modelling →