Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — simplifying an algebraic fraction
Simplify x2+7x+12x2−9.
- 1Factorise the numerator: x2−9=(x−3)(x+3)Difference of two squares — two squares with a minus between them.
- 2Factorise the denominator: two numbers multiplying to 12 and adding to 7 are 3 and 4.
- 3x2+7x+12=(x+3)(x+4)
- 4(x+3)(x+4)(x−3)(x+3)Now the common factor is visible.
- 5=x+4x−3Cancel the factor (x+3). Only whole factors cancel, never individual terms.
Example 2 — solving a quadratic by factorising
Solve x2+5x−14=0.
- 1The equation already equals zero, so factorise.The zero-product rule needs a zero on one side.
- 2Two numbers multiplying to −14 and adding to 5: that is 7 and −2.A negative product means opposite signs.
- 3(x+7)(x−2)=0
- 4So x+7=0 or x−2=0If two things multiply to zero, at least one of them is zero.
- 5x=−7 or x=2Check: 49−35−14=0 ✓ and 4+10−14=0 ✓
Example 3 — simultaneous equations by elimination
Solve 3x+2y=16 and x+y=6 simultaneously.
- 1Match the y coefficients: multiply the second equation by 2.Every term, not just the y term.
- 22x+2y=12
- 3Subtract from the first: (3x+2y)−(2x+2y)=16−12Same signs, so subtract.
- 4x=4The y terms have cancelled.
- 5Substitute into x+y=6: 4+y=6, so y=2.Both values are needed — the answer is a pair.
- 6Check in the other equation: 3(4)+2(2)=16 ✓Use the equation you did not substitute into.