Mathematics · Lesson 9

Solving quadratics

JAMBWAECNECO

What the syllabus asks

JAMB: Algebra (linear and quadratic equations, simultaneous equations)

WAEC: Algebraic processes

NECO: Algebraic processes

What you will learn

Explain what an equation is and why the "balance" rule works.
Solve linear equations, including those with brackets and fractions.
Solve simultaneous linear equations by elimination and by substitution.
Solve quadratic equations by factorisation, completing the square and the formula.
Use the discriminant to describe the roots, and use the sum and product of roots.
Turn word problems into equations, solve them, and interpret the answers.

Key points

Golden rule: do the same thing to both sides.
Linear equations: expand, clear fractions with the LCM, collect x's on one side, divide, and check.
Simultaneous equations: elimination ("Same Subtract, Different Add") or substitution.
Quadratic: ax² + bx + c = 0. Make one side zero, then use the null factor law.
Factorise using the numbers that multiply to ac and add to b.
Completing the square: add (b/2)² to both sides.
Formula: x = [−b ± √(b² − 4ac)]/2a.
Discriminant b² − 4ac: if > 0, two roots; if = 0, equal roots; if < 0, no real roots.
α + β = −b/a and αβ = c/a.
In word problems, reject impossible roots and give your reason.

Start here: What is an equation, and why does the "balance" rule work?

Let us start with a story. Mama Tolu sells tomatoes at Oja Oba. She puts a basket of tomatoes on one side of her scale and two 1 kg weights on the other side, and the scale stays level. Without opening the basket, what can you say? Of course: the basket weighs 2 kg.

That level scale is exactly what an equation is. The two sides are equal, so the scale balances. In mathematics we write this with the equals sign: basket = 2 kg.

Fig. 9.1: An equation is a scale in balance.

Definition. An equation is a mathematical statement that two expressions are equal. It always contains an equals sign (=). The letter whose value we do not know (often x) is called the unknown. To solve an equation means to find the value of the unknown that makes the statement true.

Now here is the key idea, and it is worth reading slowly. Suppose the scale is balanced, and you add a 1 kg weight to the left side only. What happens? The scale tips over, and it is no longer balanced. But if you add 1 kg to both sides at the same time, it stays balanced.

So the one golden rule of equations is:

Whatever you do to one side, you must do exactly the same to the other side.

You may add, subtract, multiply or divide both sides by the same number (except dividing by zero, which is never allowed), and the equation stays true. We use this rule to "peel away" everything around x until x is left alone on one side. Then we can read off its value.

A first example, slowly

Worked example 9.1

Solve x + 7 = 12.

We want x alone. At the moment, 7 is being added to x.
The opposite of adding 7 is subtracting 7. So we subtract 7, and because of the golden rule we do it to both sides: x + 7 − 7 = 12 − 7.
On the left, +7 − 7 cancels to nothing, leaving x. On the right, 12 − 7 = 5.
So x = 5.
Check, always: put 5 back into the original equation. 5 + 7 = 12 ✓. It balances, so we are right.

x = 5

Notice the idea of an opposite (mathematicians call it the inverse) operation. Adding is undone by subtracting, and multiplying is undone by dividing. This is like unlocking a door: if someone locked it by turning the key to the right, you open it by turning the key to the left.

If x has been…	…undo it by	Example
added to (+4)	subtracting 4 from both sides	x + 4 = 9 → x = 5
subtracted from (−4)	adding 4 to both sides	x − 4 = 9 → x = 13
multiplied (×4)	dividing both sides by 4	4x = 20 → x = 5
divided (÷4)	multiplying both sides by 4	x/4 = 3 → x = 12

Worked example 9.2

Solve x/3 = 6.

Here x has been divided by 3.
We undo division by multiplying, so we multiply both sides by 3: (x/3) × 3 = 6 × 3.
x = 18. Check: 18 ÷ 3 = 6 ✓

x = 18

Try it yourself. Solve (a) x − 9 = 4 (b) 5x = 35.

Show solution

(a) Add 9 to both sides: x = 13. (b) Divide both sides by 5: x = 7.

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Practice questions

1. Solve 7 − 2(x − 3) = x + 1.

Show worked answer

7 − 2x + 6 = x + 1, so 13 − 2x = x + 1 and 12 = 3x. x = 4.

2. Solve 3x − 2y = 4 and x + 2y = 12.

Show worked answer

Add the equations: 4x = 16, so x = 4. Then 4 + 2y = 12 gives y = 4. (4, 4)

3. Solve 6x² − x − 2 = 0.

Show worked answer

ac = −12; −4 and 3 multiply to −12 and add to −1. 6x² − 4x + 3x − 2 = 2x(3x − 2) + 1(3x − 2) = (2x + 1)(3x − 2) = 0. x = −½ or ⅔.

4. Solve x² + 6x + 2 = 0 by completing the square (surd form).

Show worked answer

x² + 6x = −2. Add 9: (x + 3)² = 7. x = −3 ± √7.

5. For what values of p does x² + px + 9 = 0 have equal roots?

Show worked answer

p² − 36 = 0, so p = ±6.

Key words

Equation: A statement that two expressions are equal, e.g. 2x + 1 = 7.
Unknown / variable: The letter whose value we are looking for (often x).
Coefficient: The number multiplying a letter. In 5x, the coefficient is 5.
Constant: A number standing alone, without a letter.
Linear: Highest power of the unknown is 1. Its graph is a straight line.
Quadratic: Highest power of the unknown is 2. Its graph is a U-shaped curve (parabola).
Root / solution: A value of the unknown that makes the equation true.
Factorise: Write an expression as a product (multiplication) of brackets.
LCM: Lowest common multiple: the smallest number that all the denominators divide into.
Discriminant: b² − 4ac. It tells you the nature of the roots.
Simultaneous equations: Two or more equations that must be true at the same time.
Surd: A root that cannot be simplified to a whole number, e.g. √5.

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