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Lesson overview

Algebra — quadratic equations

A quadratic equation has the form $ax^2 + bx + c = 0$ . **Factorising** works when the quadratic factorises into linear factors: $(px + q)(rx + s) = 0$ gives solutions $x = -\frac{q}{p}$ and $x = -\frac{s}{r}$ . **Completing the square** rewrites $x^2 + bx$ as $(x + \frac{b}{2})^2 - (\frac{b}{2})^2$ , allowing us to solve by taking square roots. **The quadratic formula** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ always works but is most efficient for equations that don't factorise easily. The discriminant $b^2 - 4ac$ tells us how many real solutions exist.

Solving by factorising

Solve $x^2 + 5x + 6 = 0$ .

Factorise the quadratic, then solve each linear factor by setting it equal to zero. This works quickly when factors exist.

Solving by completing the square

Solve $x^2 + 6x - 7 = 0$ by completing the square.

Move the constant, then complete the square on the left. Take square roots (remember ±), then solve for x.

Using the quadratic formula

Solve $2x^2 - 3x - 1 = 0$ using the quadratic formula.

Identify a, b and c. Substitute into the formula carefully. The ± gives two solutions.

Using the discriminant

How many real solutions does $x^2 - 4x + 4 = 0$ have?

If discriminant > 0: two solutions. If discriminant = 0: one solution. If discriminant < 0: no real solutions.

Worked example — choosing and comparing methods

Solve $3x^2 - 10x + 3 = 0$ using two different methods and verify the solutions.

Identify the method: this quadratic factorises, so factorising is quickest.
Factor: find two numbers that multiply to give (3)(3) = 9 and add to give −10. These are −1 and −9.
Rewrite the middle term: $3x^2 - x - 9x + 3 = 0$ .
Factor in pairs: $x(3x - 1) - 3(3x - 1) = 0$ .
Extract the common factor: $(3x - 1)(x - 3) = 0$ .
Set each factor to zero: $3x - 1 = 0$ gives $x = \frac{1}{3}$ ; $x - 3 = 0$ gives $x = 3$ .
Check: $3(\frac{1}{3})^2 - 10(\frac{1}{3}) + 3 = \frac{1}{3} - \frac{10}{3} + 3 = 0$ ✓

Try it

For each equation, decide which method is most efficient. Show your working clearly, especially when using the quadratic formula.

Question 1

Solve $x^2 - 7x + 12 = 0$ by factorising.

💡 Find two numbers that multiply to 12 and add to −7. These are −3 and −4.

Question 2

Solve $x^2 + 4x = 5$ by completing the square.

💡 Rearrange to $x^2 + 4x - 5 = 0$ . Complete the square: $(x + 2)^2 - 4 - 5 = 0$ .

Question 3

Use the quadratic formula to solve $2x^2 - 5x + 2 = 0$ .

💡 Substitute $a = 2$ , $b = -5$ , $c = 2$ into $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Question 4

How many real solutions does $x^2 - 2x + 1 = 0$ have?

💡 Calculate the discriminant: $b^2 - 4ac = 4 - 4 = 0$ .

Question 5

Solve $x^2 + 2x + 5 = 0$ .

💡 The discriminant is $4 - 20 = -16 < 0$ , so there are no real solutions.

Common mistakes

Watch for these when working through the lesson.

When factorising, forgetting to extract the common factor fully, or missing a middle term split.
In the quadratic formula, making sign errors with b and c. Write it out clearly: $-b$ is the opposite of b.
When completing the square, forgetting to move (½b ${)}^{2}$ to both sides, or using it incorrectly.
Not checking whether solutions are reasonable by substituting back into the original equation.

Solve quadratics by factorising, completing the square and formula

Lesson overview

Worked example — choosing and comparing methods

Try it

Common mistakes

Related topics

Independent practice will plug in here