Quadratic equations

• Always rearrange to ax² + bx + c = 0 before you start.
• Factorise: if (x + p)(x + q) = 0 then x = −p or x = −q.
• Higher: factorise ax² + bx + c; use the quadratic formula (given) or complete the square.
• Higher: form a quadratic from a context and reject solutions that make no sense.
• Higher: solve a linear and a quadratic equation simultaneously by substitution.

x² + bx + c = 0

If two numbers multiply to give 0, one of them must be 0. So after factorising, set each bracket equal to zero.

ax² + bx + c = 0

Rearrange so one side is zero, then factorise (multiply to ac, add to b). A bracket like (3x − 5) = 0 gives a fractional solution, x = 5/3.

When it will not factorise

If the question asks for answers to a number of decimal places or significant figures, it will usually not factorise: use the formula. If it asks for an exact answer in surd form, completing the square is often neater.

Form, solve, then interpret

Use the information to write an equation, rearrange it to = 0, solve, and then reject any solution that is impossible (a negative length, a fraction of a person). Say why you rejected it.

Substitute the linear into the quadratic

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