Expand and factorise simple quadratics

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Expand and factorise simple quadratics

When two binomials multiply, every term in one bracket multiplies every term in the other. For (x+2)(x+3), the four products are x², 3x, 2x and 6. Combining the middle terms gives x²+5x+6.

To reverse this for x²+bx+c, seek two numbers whose sum is b and product is c. Their signs determine both conditions. Checking only the product can give a factorisation with the wrong linear coefficient.

The difference of two squares is a useful special identity: a²−b²=(a−b)(a+b). The middle terms cancel during expansion. A sum of two squares does not follow that same real-factor pattern.

Worked example: Factorise x²+x−6.

Find numbers with product−6 and sum 1:3 and −2.

Use the brackets(x+3)(x−2).

Expand to check x²−2x+3x−6.

Answer: (x+3)(x−2).

Check your understanding: Factorise x²−25.

Answer: (x−5)(x+5).

Why: It is a difference of squares, with 25=5².