How to Use the Quadratic Formula — Step by Step

If you're preparing for Grade 12 maths — whether NSC or IEB — the quadratic formula is one of the most important tools you'll ever use. Every year, it appears in Paper 1. Every year, students lose marks because of small, avoidable errors. This guide will fix that.

We're not just showing you the formula. We're showing you how to think through it, step by step, so it becomes automatic under exam pressure.

The Quadratic Formula (And What Each Part Means)

The formula solves any equation of the form $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Before you plug in a single number, you need to know what you're working with:

• a is the coefficient of $x^2$ (the number in front of x-squared)
• b is the coefficient of $x$ (the number in front of x)
• c is the constant (the number on its own)
• $\Delta = b^2 - 4ac$ is the discriminant — it tells you how many solutions exist before you calculate them

Getting comfortable identifying a, b, and c correctly is the single biggest factor in using this formula without errors.

The Discriminant First — Always

Most students skip straight to substituting into the formula. Don't. Calculate the discriminant ($\Delta = b^2 - 4ac$) first and use it to predict your answer:

| $\Delta$ | What it means | How many real roots | |---|---|---| | $\Delta > 0$ | Two distinct real roots | 2 | | $\Delta = 0$ | One repeated root (equal roots) | 1 | | $\Delta < 0$ | No real roots | 0 |

This step takes 30 seconds and prevents the nasty surprise of getting $\sqrt{\text{negative number}}$ halfway through a test.

The 5-Step Method

Apply these steps to every quadratic formula problem, without exception:

1. Rearrange — get the equation into $ax^2 + bx + c = 0$ form
2. Identify $a$, $b$, and $c$ (write them down explicitly)
3. Calculate $\Delta = b^2 - 4ac$ (determine nature of roots)
4. Substitute into the formula
5. Simplify — leave surds in simplest form, or round to the specified decimal places

Never skip Step 1. If the equation isn't equal to zero, you'll identify the wrong values and everything falls apart.

Worked Example 1: Straightforward Substitution

Solve: $x^2 + 7x + 10 = 0$

(Note: this one factors easily — I'm starting here so you can verify the formula gives the same answer.)

Step 1: Already in standard form. ✓

Step 2: $a = 1$, $b = 7$, $c = 10$

Step 3: $\Delta = (7)^2 - 4(1)(10) = 49 - 40 = 9$ → Two distinct real roots ✓

Step 4:

$x = \frac{-7 \pm \sqrt{9}}{2(1)} = \frac{-7 \pm 3}{2}$

Step 5:

$x = \frac{-7 + 3}{2} = \frac{-4}{2} = \mathbf{-2}$

$x = \frac{-7 - 3}{2} = \frac{-10}{2} = \mathbf{-5}$

Check: Factored form is $(x + 2)(x + 5) = 0$. Same answers. ✓

Worked Example 2: Irrational Roots (Surds)

Solve: $2x^2 - 5x + 1 = 0$, leaving answers in simplest surd form

Step 2: $a = 2$, $b = -5$, $c = 1$

Step 3: $\Delta = (-5)^2 - 4(2)(1) = 25 - 8 = 17$

17 is prime, so $\sqrt{17}$ cannot be simplified. Two irrational roots.

Step 4:

$x = \frac{-(-5) \pm \sqrt{17}}{2(2)} = \frac{5 \pm \sqrt{17}}{4}$

Step 5: This is already fully simplified.

$x = \frac{5 + \sqrt{17}}{4} \quad \text{or} \quad x = \frac{5 - \sqrt{17}}{4}$

This is your final answer. Do not convert to decimals unless specifically asked.

Worked Example 3: Must Rearrange First

Solve: $3x^2 = 4x + 7$

This is a common exam trap. The equation is NOT in standard form.

Step 1: $3x^2 - 4x - 7 = 0$ (subtract $4x$ and $7$ from both sides)

Step 2: $a = 3$, $b = -4$, $c = -7$

Step 3: $\Delta = (-4)^2 - 4(3)(-7) = 16 + 84 = 100$

$\sqrt{100} = 10$. Exact roots expected.

Step 4:

$x = \frac{-(-4) \pm 10}{2(3)} = \frac{4 \pm 10}{6}$

Step 5:

$x = \frac{4 + 10}{6} = \frac{14}{6} = \mathbf{\frac{7}{3}}$

$x = \frac{4 - 10}{6} = \frac{-6}{6} = \mathbf{-1}$

Worked Example 4: Decimal Answers

Solve: $x^2 - 3x - 5 = 0$, giving answers correct to two decimal places

Step 2: $a = 1$, $b = -3$, $c = -5$

Step 3: $\Delta = 9 + 20 = 29$ → two irrational roots

Step 4:

$x = \frac{3 \pm \sqrt{29}}{2}$

Step 5:

$\sqrt{29} \approx 5.385$

$x = \frac{3 + 5.385}{2} = \frac{8.385}{2} \approx \mathbf{4.19}$

$x = \frac{3 - 5.385}{2} = \frac{-2.385}{2} \approx \mathbf{-1.19}$

Crucial habit: carry at least one more decimal place in intermediate working (use 5.385), then round only at the final step.

Worked Example 5: No Real Solution

Solve: $x^2 + 2x + 5 = 0$

Step 2: $a = 1$, $b = 2$, $c = 5$

Step 3: $\Delta = 4 - 20 = -16$

Since $\Delta < 0$, there are no real roots. Stop here. Write: "No real solution."

In the CAPS curriculum, you are not expected to work with complex numbers at school level. The correct answer is to state there are no real roots. A common mistake is pushing through and getting $\sqrt{-16}$ — which costs marks.

The Five Most Common Mistakes

After marking hundreds of Grade 12 assessments, here are the errors I see most often:

1. Forgetting to rearrange to $ax^2 + bx + c = 0$ first. If there's a term on the right-hand side, move it before identifying $a$, $b$, $c$.

2. Dropping the negative sign when $b$ is negative. If $b = -5$, then $-b = +5$. This is where many students go wrong.

3. Dividing by $2a$ too early. The entire numerator $(-b \pm \sqrt{\Delta})$ divides by $2a$ — not just the $\pm\sqrt{\Delta}$ part. Use brackets.

4. Not simplifying surds. $\sqrt{12}$ should be simplified to $2\sqrt{3}$. Examiners deduct marks for unsimplified answers.

5. Rounding too early. Keep exact values or extra decimal places until the very last step.

What the Examiner Is Looking For

In CAPS Paper 1, a typical quadratic formula question is worth 4–6 marks. Marks are allocated approximately like this:

• Correct standard form: 1 mark
• Correct substitution into formula: 1 mark
• Correct simplification / discriminant: 1 mark
• Correct final answers (both roots): 2 marks

Even if you can't reach the final answer, you can earn 3 of those 4 marks by setting up correctly. Never leave a quadratic question blank.

Practice Problems

Work through these before your next test — the more you practise, the more automatic the steps become:

1. $x^2 - 9x + 20 = 0$ (should factor — use formula to verify)
2. $3x^2 + 2x - 8 = 0$ (leave in fraction form if exact)
3. $x^2 = 6x - 1$ (rearrange first, surd answer)
4. $2x^2 + x + 3 = 0$ (no real solution — prove it)
5. $5x^2 - 3 = 7x$ (decimal, 2dp)

Check your answers by substituting back into the original equation.

Need More Help?

Quadratic equations are a gateway topic in Grade 12 maths — they appear in functions, calculus, and algebraic reasoning. If the formula still feels shaky, or if you want to work through exam-style questions with immediate feedback, book a session with me. I teach maths from Grade 8 through matric, in English and Afrikaans.

You can also find more maths resources on the resources page, or see how exam prep sessions work if you're heading into a big test.