Quadratic equations: three mistakes worth checking

Quadratic equations are one of the most fundamental topics in GCSE and A-level maths, yet many students struggle with them. Here are three mistakes worth looking for, with examples of how to check the reasoning.

Three common mistakes

1. Forgetting the zero-product rule

Many students forget that when you factor a quadratic equation and get something like $(x + 3)(x - 5) = 0$, you need to set EACH factor equal to zero separately.

The mistake: Student works: $x(x + 3) = 0$ Student answer: $x = -3$ (missing the second solution!)

The correct approach: When $x(x + 3) = 0$, then either $x = 0$ OR $x + 3 = 0$

Therefore: $x = 0$ or $x = -3$

The tip: After factoring, ask yourself “What value makes this bracket equal to zero?” - do this for EACH bracket!

2. Choosing a factorisation method

Students often confuse when to use:

• Factorizing (when there are two terms with a common factor)
• The difference of two squares
• Complete factorization

Quick guide:

• Two terms with common factor: Factor it out
• A difference of two perfect squares: Use the difference-of-squares identity
• Three terms (ax² + bx + c): Factorize or use the quadratic formula

3. Sign errors with negative values

This is subtle but critical. When working with equations like $x^2 - 4x - 5 = 0$, students often forget to check if their factors can actually multiply to give -5 (which requires one positive and one negative factor).

The tip: Always check your factors work! Multiply them back out to verify.

A worked example

For $x^2 - 6x + 8 = 0$, we need factors of 8 that add to -6. Two tempting choices are:

• +8 and +1 (adds to 9, too high)
• +4 and +2 (adds to 6, but wrong sign)

The insight: When the middle term is negative and the last term is positive, BOTH factors must be negative!

The correct factors: $(x - 2)(x - 4) = 0$, giving $x = 2$ or $x = 4$.

Checks worth building into your working

1. Always check your answers: Plug your solutions back into the original equation
2. Practice spotting patterns: Difference of two squares shows up frequently in GCSE
3. Don’t jump to the formula: Try factoring first - it’s faster and builds understanding

Remember how many roots are possible

The biggest “aha moment” I see is when students realize that quadratic equations have UP TO two solutions. Once they understand this, they start checking whether they’ve found both solutions, not just one.

Use the discriminant to check the real roots

For a quadratic $ax^2 + bx + c = 0$, with $a \ne 0$, the discriminant helps identify the real solutions:

$$\Delta = b^2 - 4ac$$

This tells you:

• If $\Delta < 0$: No real solutions
• If $\Delta = 0$: One repeated real solution
• If $\Delta > 0$: Two distinct real solutions

For example, with $x^2 - 6x + 8 = 0$:

$$\Delta = (-6)^2 - 4(1)(8) = 36 - 32 = 4 > 0$$

So we know there are two solutions before we even factor!

What to practise next

If you’re struggling with quadratic equations, start by practicing these three types:

1. When the constant term is zero (e.g., $x^2 - 5x = 0$)
2. When it’s a perfect square trinomial (e.g., $x^2 - 6x + 9 = 0$)
3. The general case with two different solutions

Remember: mastering quadratics opens the door to so much more in mathematics - from trigonometry to calculus. Take your time with the basics!