Solving Quadratics
Solving quadratics means finding the value or values of x that make an equation of the form ax^2 + bx + c = 0 true. GCSE questions expect solutions by factorising where possible, and by using the quadratic formula when a quadratic will not factorise neatly with whole numbers.
Before you start
Make sure you're comfortable with these topics first:
Method
- Rearrange the equation so that one side equals zero, with all terms in the form ax^2 + bx + c = 0.
- Try to factorise the quadratic into two brackets; if it factorises, set each bracket equal to zero and solve each mini-equation separately.
- If the equation only has an x^2 term and a number (no x term), rearrange to x^2 = number and take the square root of both sides, remembering both the positive and negative roots.
- If the equation will not factorise with whole numbers, use the quadratic formula x = (-b +- sqrt(b^2 - 4ac)) / (2a), substituting a, b and c carefully, including their signs.
- Give both solutions unless the context of the question rules one out, for example if x represents a length, reject any negative solution.
- Check each solution by substituting it back into the original equation.
Worked example
Solve x^2 - 2x - 24 = 0.
- Find two numbers that multiply to give -24 and add to give -2: these are -6 and 4.
- Factorise: (x - 6)(x + 4) = 0.
- Set each bracket equal to zero: x - 6 = 0 or x + 4 = 0.
- Solve each equation: x = 6 or x = -4.
- State the final answer: x = 6 or x = -4.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Solve x^2 - 3x - 18 = 0.
Solve x^2 - 4x - 3 = 0. Use the quadratic formula and give your answers correct to 2 decimal places.
A right-angled triangle has sides x cm, (x + 2) cm and (x + 4) cm, where (x + 4) cm is the hypotenuse. (a) Show that x^2 - 4x - 12 = 0. (3) (b) Solve the equation to find x, and hence find the perimeter of the triangle. (3)
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Free printable worksheet
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