Reasoning with Equations and Inequalities
Quadratic equations
Solve quadratic equations by factorizing, and use the discriminant to describe the roots.
A quadratic equation is one where the highest power of the unknown is 2. That single fact changes everything: a linear equation has one solution, but a quadratic can have two, and both of them are equally real answers to the question.
Why factorizing works
If two things multiply to give zero, at least one of them must be zero — nothing else can produce a zero product. So once you have written the equation as (x + p)(x + q) = 0, you only have to ask what makes each bracket vanish.
Worked example
Solve x² - 5x + 6 = 0.
Look for two numbers that multiply to 6 and add to -5. They are -2 and -3.
When you expand (x + p)(x + q) you get x² + (p + q)x + pq — so p and q must add to the middle number and multiply to the last one.
Try it together
Your turn: solve x² + 7x + 12 = 0.
Same hunt as before — two numbers that multiply to 12 and add to 7.
1.The pairs that multiply to 12 are 1 and 12, 2 and 6, 3 and 4. Which pair also adds to 7? Type the smaller of the two numbers.
When it will not factorize
The discriminant b² - 4ac tells you what to expect before you start. Positive means two different roots, zero means the two roots are the same, and negative means no real roots at all — no pair of whole numbers will ever work, because there is nothing to find.
Have a go
Have a go on your own: solve x² - 6x + 8 = 0 and give the larger root.
Hint: Two numbers that multiply to 8 and add to -6.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.