lesson
Find two numbers with a product and a sum
1 · Learn
To factor x²+bx+c, find numbers p and q whose product is c and whose sum is b. Then x²+bx+c=(x+p)(x+q). Include negative factor pairs when c is positive or negative.
To solve a factorised quadratic equal to zero, use the zero-product property: if (x+p)(x+q)=0, then x=-p or x=-q. Factorising an expression and solving an equation are related but different tasks.
Expand the factors to verify every coefficient. A missing solution often comes from setting only one factor to zero.
2 · Worked example
x²+x-12 uses 4 and -3: (x+4)(x-3). Therefore x²+x-12=0 has x=-4 or x=3.
3 · Your turn
Factorise x²+7x+12, then solve x²+7x+12=0.
Check your answer
(x+3)(x+4); x=-3 or x=-4.
4 · Apply your learning
Make a factor-pair table for c. Circle the pair whose sum is b, then verify by expansion and substitution.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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