The formula and the discriminant · Quadratics & their graphs

Completing the square on $ax^2 + bx + c = 0$ with letters instead of numbers gives one formula that solves every quadratic.

What the formula means on the graph

The formula has two parts. $-\frac{b}{2a}$ is the line of symmetry. The $\pm$ part is how far each root sits from that line: one root adds the distance, the other takes it away.

The formula on the graphMove the sliders

y = x² − 6x + 5

x = (−b ± √(b² − 4ac)) ÷ 2a

= (6 ± √((−6)² − 4 × 1 × 5)) ÷ 2

= (6 ± √16) ÷ 2

= 1 or 5

16discriminant b² − 4actworeal rootsx = 3line of symmetry, −b ÷ 2a

a = 1b = −6c = 5

The two roots sit either side of the line x = 3, each √(b² − 4ac) ÷ 2a = 2 away. That is what the ± in the formula does: one root adds the distance, the other takes it away.

Move $c$ slowly upwards (with $a = 1$ ) and watch the two roots close in, meet, and then disappear. Watch the number under the square root as you do it.

The discriminant

The part under the square root, $b^2 - 4ac$ , is called the discriminant. It tells you how many roots there are before you do any other working.

$b^2 - 4ac$	Roots	The graph
positive	two different real roots	crosses the $x$ -axis twice
zero	one repeated root (equal roots)	touches the $x$ -axis
negative	no real roots	misses the $x$ -axis

“Equal roots” questions use the discriminant as an equation. For $x^2 + kx + 9 = 0$ to have equal roots, $k^2 - 4 \times 1 \times 9 = 0$ , so $k^2 = 36$ and $k = \pm 6$ .

More: the discriminant and the formula

Using the formula carefully

Write down $a$ , $b$ and $c$ with their signs before you substitute.
Work out $b^2 - 4ac$ on its own first.
Put brackets round negative numbers: $(-3)^2 = 9$ , while $-3^2 = -9$ .

One linear, one quadratic

When one equation is linear and the other quadratic, make one letter the subject of the linear equation and substitute into the quadratic. That leaves a quadratic in one letter: solve it, then use the linear equation to find the other letter for each answer. On a graph, the answers are where the line meets the curve.

A line and a curveThe two crossing points are the two solutions

More: one linear, one quadratic

Word problems

Many marks are lost on word problems because the equation never gets written down. Follow these steps:

Choose a letter for the unknown and say what it stands for (“let the width be $w$ cm”).
Write the other quantities in terms of it (the length is $w + 3$ ).
Form the equation from the fact you’re given (the area is 70, so $w(w + 3) = 70$ ).
Rearrange to $= 0$ and solve.
Check the answers make sense. A length or an age can’t be negative, so reject that root.

A past question, step by step

Worked example · NECO 2024

NECO 2024 · Paper 1 · Q33

The difference between the present ages of two brothers is 6 and their product is 135. What is the sum of their ages?

Choose a letter
Let the younger brother be $x$ years old.
Think first. If the younger brother is $x$ years old, how old is the elder?
Form the equation
The elder is $x + 6$ , and the product of their ages is 135:
$x(x + 6) = 135 \quad\Rightarrow\quad x^2 + 6x - 135 = 0$
Think first. Rearrange so one side is 0. Which two numbers multiply to $-135$ and add to $6$ ?
Solve
$15 \times (-9) = -135$ and $15 - 9 = 6$ :
$(x + 15)(x - 9) = 0 \quad\Rightarrow\quad x = 9 \text{ or } x = -15$
Check it makes sense
An age can’t be negative, so $x = 9$ . The brothers are 9 and 15, and the sum of their ages is $24$ . The answer is B.

Your turn

WAEC 2015 · Paper 2 · Q2 (b)

(b)
The diagram shows a rectangle $PQRS$ , $20\text{ cm}$ high, from which a square of side $x\text{ cm}$ has been cut out of the middle of the base, leaving $10\text{ cm}$ on each side. If the area of the shaded portion is $484\text{ cm}^2$ , find the values of $x$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(b)

The rectangle is 20 cm high and $10 + x + 10 = 20 + x$ cm wide.
The shaded area is the rectangle without the square: $20(20 + x) - x^2 = 484$ .
Expand: $400 + 20x - x^2 = 484$ , so $x^2 - 20x + 84 = 0$ .
Factorise: $(x - 6)(x - 14) = 0$ , so $x = 6$ or $x = 14$ .

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