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Lesson overview

Algebra — equations and manipulation

Advanced algebra at GCSE includes solving quadratics by multiple methods (factorising, completing the square, formula), manipulating simultaneous equations (including one linear and one quadratic), working with algebraic fractions, and rearranging complex formulas. Fluency in these skills is essential because exam questions often embed them within larger problems. Common fragile areas include: choosing the best method for a given quadratic, avoiding arithmetic errors in the formula, and checking solutions.

Solving a quadratic by formula

Solve $2x^2 - 5x + 2 = 0$ using the quadratic formula.

Always identify a, b, c carefully, especially if b or c is negative.

Simultaneous equations: one linear, one quadratic

Solve: $y = 2x + 1$ and $y = x^2 - 4x + 3$ .

Substitute one equation into the other, then rearrange and solve.

Rearranging a formula with fractions

Rearrange $v = u + at$ to make $t$ the subject.

Perform inverse operations to isolate the target variable. Work step by step.

Factorising a quadratic with coefficient > 1

Factorise $3x^2 - 11x + 6$ .

For ${ax}^{2}$ + bx + c, find factors of ac that add to b, then use grouping.

Worked example — multi-step algebraic problem

Solve $5x^2 - 3x - 2 = 0$ and verify your answer by substitution.

Identify a = 5, b = −3, c = −2.
Calculate the discriminant: ${b}^{2}$ − 4ac = (−3 ${)}^{2}$ − 4(5)(−2) = 9 + 40 = 49.
Since $\sqrt{49}$ = 7, use the formula: x = (3 ± 7) / 10.
Solve: x = $\frac{10}{10}$ = 1 or x = − $\frac{4}{10}$ = −0.4.
Verify x = 1: 5(1 ${)}^{2}$ − 3(1) − 2 = 5 − 3 − 2 = 0 ✓
Verify x = −0.4: 5(0.16) − 3(−0.4) − 2 = 0.8 + 1.2 − 2 = 0 ✓

Try it

Use the examples carefully, then choose the answer.

Question 1

Factorise $x^2 - 7x + 12$ .

💡 Find two numbers that multiply to 12 and add to −7.

Question 2

Solve $x^2 + 3x - 10 = 0$ using the quadratic formula.

💡 a = 1, b = 3, c = −10. Calculate ${b}^{2}$ − 4ac = 9 + 40 = 49.

Question 3

If y = 3x − 1 and y = ${x}^{2}$ + 2x − 3, find the value(s) of x.

💡 Substitute: 3x − 1 = ${x}^{2}$ + 2x − 3. Rearrange to ${x}^{2}$ − x − 2 = 0.

Question 4

Rearrange T = 2π $\sqrt{L/g}$ to make L the subject.

💡 Square both sides, then rearrange. Be careful with the denominator.

Question 5

Solve 2 ${x}^{2}$ − 7x + 3 = 0. Enter the larger solution (to 2 d.p.).

💡 Use the formula or factorise: (2x − 1)(x − 3) = 0.

Common mistakes

Watch for these when working through the lesson.

Getting signs wrong in the quadratic formula, especially with negative coefficients.
Forgetting to divide by 2a in the formula (using only −b ± $\sqrt{b^{2} − 4ac}$ ).
Not simplifying before using the formula (e.g., dividing through by a common factor first).
Losing solutions when rearranging simultaneous equations; always check that both solutions satisfy both original equations.

Advanced algebra revision

Lesson overview

Worked example — multi-step algebraic problem

Try it

Common mistakes

Related topics

Independent practice will plug in here