Solving Quadratic Equations by Factorising
Rearrange a quadratic equation so one side is zero, factorise it, then use the fact that a product is zero only when one factor is zero to read off both solutions.
What a learner can do afterwards
- Rearrange x² + 5x = 14 into x² + 5x - 14 = 0 before factorising
- Apply the zero product rule to get both solutions from (x - 2)(x + 7) = 0
- Reject a solution that cannot fit the context, such as a negative length
1 · Read
The last three stops gave you a factorising toolkit: the pair method, the special patterns, and splitting the middle term. This stop turns factorising into a way to solve equations. The key is a small fact: a product is zero only when one of its factors is zero. If you can write the left side as two brackets, each bracket hands you a solution.
Here is the rule in full. If A times B equals zero, then A is zero or B is zero. There is no other way for a product to be zero. The common slip is to set the two factors equal to each other, like writing x - 3 = x + 4 from (x - 3)(x + 4) = 0. The equation says the product is zero, so each factor gets its own equation: x - 3 = 0 and x + 4 = 0.
Solve x squared + 3x - 10 = 0. The equation is already set to zero, so factorise straight away. The pair that multiplies to -10 and adds to 3 is 5 and -2, so the left side is (x + 5)(x - 2). Now the zero product rule: x + 5 = 0 or x - 2 = 0, giving x = -5 or x = 2. Check both in the original: (-5) squared + 3(-5) - 10 is 25 - 15 - 10 = 0, and 2 squared + 3(2) - 10 is 4 + 6 - 10 = 0.
Now an equation that is not yet set to zero. Solve 3x squared = 12x. Move every term to one side: 3x squared - 12x = 0. Take out the common factor 3x: 3x(x - 4) = 0. The rule gives x = 0 or x = 4. Notice why you must not divide both sides by x: dividing quietly drops the solution x = 0, because it assumes x is not zero.
Factorising also works when the answer has to make sense. A garden has width x metres and length (x + 3) metres, and its area is 40 square metres. So x(x + 3) = 40. Expand and set to zero: x squared + 3x - 40 = 0. The pair that multiplies to -40 and adds to 3 is 8 and -5, so (x + 8)(x - 5) = 0, giving x = -8 or x = 5. A width of -8 metres is impossible, so x = 5: the garden is 5 by 8 metres.
The two solutions can look very different: one positive and one negative, one a fraction and one a whole number. That is normal. Trust the rule, write down both, and only then check them against the original equation or the context.
Move every term to one side so the other is zero, factorise, set each factor to zero, and keep the solutions that fit the context.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.