Solving Quadratics by Factorisation: Method & Examples

Solving a quadratic by factorisation uses one simple fact: if the product of two numbers is zero, then at least one of them is zero. This is the zero product property. So if you can rewrite ax squared + bx + c = 0 as (px + q)(rx + s) = 0, the solutions are just x = -q/p and x = -s/r.

This matters because factorisation is by far the fastest way to solve a quadratic when it works, and exam questions are often designed for it. It also builds the number sense that makes the quadratic formula feel natural later.

Key Concepts

• Zero product property: if A times B = 0 then A = 0 or B = 0 (or both). This only works when the product equals zero.
• Product-sum rule: for x squared + bx + c, find two numbers with product c and sum b; the factors are (x + m)(x + n).
• Splitting the middle term: for non-monic quadratics, split bx using two numbers with product a times c and sum b, then group.
• Standard form first: everything must be on one side with zero on the other before the zero product property can be used.

How It Works

1. Move every term to one side so the equation reads (quadratic) = 0.
2. Factorise the quadratic: use the product-sum rule if the leading coefficient is 1, otherwise split the middle term and group.
3. Set each linear factor equal to zero separately.
4. Solve each tiny linear equation to get the roots.
5. Substitute each root back into the original equation to verify.

Worked Example

Solve x squared + 5x + 6 = 0.

1. The equation is already in standard form with zero on the right.
2. Find two numbers with product 6 and sum 5: they are 2 and 3.
3. Factorise: (x + 2)(x + 3) = 0.
4. Set each factor to zero: x + 2 = 0 gives x = -2; x + 3 = 0 gives x = -3.
5. Solutions: x = -2 or x = -3.

Check x = -2: 4 - 10 + 6 = 0. Check x = -3: 9 - 15 + 6 = 0. Both satisfy the equation.

Common Misconceptions

• You can set each factor to zero even if the right side is not zero. Wrong. The zero product property needs the product to equal zero. First rearrange, for example x squared + 5x = -6 must become x squared + 5x + 6 = 0.
• (x + 2)(x + 3) = 0 means x = 2 or x = 3. Wrong. Solving x + 2 = 0 gives x = -2, not 2. Watch the signs.
• If it does not factorise quickly, factorisation has failed. Not failed, just unsuitable: some quadratics have no integer factors, and then the quadratic formula is the right tool.

Real-World Applications

• In physics, finding when a projectile hits the ground means solving a quadratic, and factorisation gives the time directly.
• In geometry, rectangle problems where area and perimeter are known lead to quadratics solved by factorisation.
• In number puzzles, finding two integers from their sum and product is exactly the product-sum step of this method.

To recap: get zero on one side, factorise, set each factor to zero, and solve. Never apply the zero product property before rearranging, keep track of signs, and verify each root by substitution.