Quadratic Expressions
This topic covers 5 learning steps, guiding your child from the basics through to confident problem-solving. Each step includes a worked example and adaptive practice questions.
What Your Child Will Learn
- Solve by factorising
Solve x² + 5x + 6 = 0 - The quadratic formula
Use the formula for any quadratic - Discriminant
Determine number of solutions from b²-4ac - Completing the square
Rewrite in (x + p)² + q form - Word problems
Form and solve quadratics from context
Common Mistakes
- Thinking that x² = 9 only has one solution (x = 3), forgetting the negative root (x = -3)
Every positive number has TWO square roots: a positive one and a negative one. x² = 9 gives x = 3 AND x = -3, because (-3) × (-3) = 9 too. Only x² = 0 has a single solution. - Incorrectly factorising by getting the signs wrong (e.g. factorising x² - x - 6 as (x - 3)(x + 2) instead of (x - 3)(x + 2) but then solving wrong — setting each bracket to zero incorrectly)
After factorising, set each bracket equal to zero separately. For (x - 3)(x + 2) = 0: either x - 3 = 0 so x = 3, or x + 2 = 0 so x = -2. Check by expanding: (x - 3)(x + 2) = x² + 2x - 3x - 6 = x² - x - 6.
Tips for Parents
- Think of quadratics as equations with x² in them. The graph is always a U-shape (or upside-down U). The solutions are where the curve crosses the x-axis.
- Practise factorising by finding two numbers that multiply to give the last number and add to give the middle number. For x² + 5x + 6: which two numbers multiply to 6 and add to 5? Answer: 2 and 3.
- The quadratic formula is a safety net that always works: x = (-b ± √(b² - 4ac)) / 2a. Help your child substitute values carefully, one step at a time.
- Use the discriminant (b² - 4ac) as a quick check: if it is positive, there are 2 solutions; if it is zero, there is 1 solution; if it is negative, there are no real solutions.
Key Words
- Quadratic — An expression or equation where the highest power of x is 2 — like x² + 3x - 10.
- Factorise — Write as a product of brackets — x² + 5x + 6 = (x + 2)(x + 3).
- Root (or solution) — A value of x that makes the equation equal zero — the roots of x² - 9 = 0 are x = 3 and x = -3.
- Discriminant — The value b² - 4ac, which tells you how many solutions a quadratic equation has.
- Completing the square — Rewriting a quadratic in the form (x + p)² + q — useful for finding the turning point of a parabola.
- Parabola — The U-shaped curve you get when you plot a quadratic equation.
Where This Fits
Before this topic: Children should be confident expanding and factorising single and double brackets, and understand square numbers and square roots.
After this topic: Quadratics lead to sketching parabolas, solving quadratic inequalities, quadratic simultaneous equations, and the quadratic formula at GCSE.
How MathCraft Teaches This
In MathCraft, Quadratic Expressions is taught through the Algebra & Arithmetic adventure track. Your child follows guided lessons with friendly characters, works through examples step by step, then practises with questions that adapt to their level.
The adaptive engine tracks mastery across all 5 steps, revisiting concepts your child finds tricky and advancing when they're ready. Parents can see detailed progress in the Parent Dashboard.
Practise Quadratic Expressions with MathCraft
Step-by-step lessons, worked examples, and adaptive practice — all wrapped in an adventure game your child will love.
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