How to Solve Quadratic Equations — Complete Guide

A quadratic equation is any equation that can be written in the form $ax^2 + bx + c = 0$, where a, b, and c are constants and $a \neq 0$. These equations appear every year in NSC and IEB matric Paper 1 — in algebra, functions, calculus optimisation, and financial maths. You need to be fluent with all three solution methods and know which one to reach for first.

This guide takes you from the basics through to exam-level technique, with worked examples at each stage.

The Three Methods — When to Use Each

| Method | Use when… | |---|---| | Factoring | The equation factors neatly (trinomials with integer roots) | | Completing the square | Asked explicitly, or deriving the vertex of a parabola | | Quadratic formula | Always works — use when factoring isn't obvious |

A common exam error is spending three minutes trying to factor an equation that doesn't factor. If you don't spot factors within 60 seconds, move straight to the formula.

Before You Start: Standard Form

Every solution method requires the equation to be in $ax^2 + bx + c = 0$ form first. If you're given something like $3x^2 = 4x + 7$, you must rearrange before doing anything else.

Failing to rearrange first is the single most common reason students identify a, b, and c incorrectly — and then get both roots wrong.

Method 1: Factoring

Factoring works by rewriting the quadratic as a product of two linear factors, then applying the zero product property: if $A \times B = 0$, then $A = 0$ or $B = 0$.

How to Factor $ax^2 + bx + c$

1. Find two numbers that multiply to $a \times c$ and add to b
2. Split the middle term and factor by grouping (or write factors directly if $a = 1$)

Worked Example 1: a = 1

Solve: $x^2 + 7x + 12 = 0$

We need two numbers that multiply to 12 and add to 7: those are 3 and 4.

$$(x + 3)(x + 4) = 0$$ $$x = -3 \quad \text{or} \quad x = -4$$

Check: $(-3)^2 + 7(-3) + 12 = 9 - 21 + 12 = 0$ ✓

Worked Example 2: $a \neq 1$

Solve: $6x^2 - x - 2 = 0$

We need two numbers that multiply to $6 \times (-2) = -12$ and add to $-1$: those are −4 and 3.

$6x^2 - 4x + 3x - 2 = 0$

$2x(3x - 2) + 1(3x - 2) = 0$

$(2x + 1)(3x - 2) = 0$

$x = -\frac{1}{2}$ or $x = \frac{2}{3}$

When Factoring Fails

Not every quadratic factors over the integers. If you can't find whole-number factors, the roots are irrational and you need the quadratic formula. Don't force it.

Method 2: Completing the Square

This method rewrites $ax^2 + bx + c = 0$ by creating a perfect square trinomial on the left side. It's less common in routine solving, but examiners ask for it specifically when deriving formulas or finding turning points.

The Steps

1. Divide everything by $a$ so the $x^2$ term has coefficient 1
2. Move the constant to the right: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
3. Add $\left(\frac{1}{2} \times \text{coefficient of } x\right)^2$ to both sides
4. Write the left side as a perfect square
5. Solve by taking square roots of both sides

Worked Example 3: Completing the Square

Solve: $x^2 + 6x - 7 = 0$

Step 1: Coefficient of $x^2$ is already 1.

Step 2: $x^2 + 6x = 7$

Step 3: Half of 6 is 3; $3^2 = 9$. Add 9 to both sides:

$x^2 + 6x + 9 = 16$

Step 4: $(x + 3)^2 = 16$

Step 5: $x + 3 = \pm 4$

$x = 1 \quad \text{or} \quad x = -7$

Check (by factoring): $(x - 1)(x + 7) = 0$ → same answers ✓

Worked Example 4: Completing the Square with Surds

Solve: $x^2 - 4x - 1 = 0$, leaving answers in simplest surd form

$x^2 - 4x = 1$

$x^2 - 4x + 4 = 5$

$(x - 2)^2 = 5$

$x - 2 = \pm\sqrt{5}$

$x = 2 + \sqrt{5} \quad \text{or} \quad x = 2 - \sqrt{5}$

These cannot be simplified further. Never convert to decimals unless the question specifically asks.

Method 3: The Quadratic Formula

The formula solves any quadratic equation:

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

Always Calculate the Discriminant First

The discriminant $\Delta = b^2 - 4ac$ tells you how many real solutions exist before you calculate them:

| $\Delta$ | Nature of roots | |---|---| | $\Delta > 0$ | Two distinct real roots | | $\Delta = 0$ | One repeated real root | | $\Delta < 0$ | No real roots |

This 30-second step prevents the surprise of getting $\sqrt{\text{negative number}}$ mid-calculation.

Worked Example 5: Standard Formula Use

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

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

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

$x = \frac{5 \pm \sqrt{17}}{4}$

Since 17 is prime, $\sqrt{17}$ cannot be simplified. This is the final answer.

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

Worked Example 6: Decimal Answer

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

$a = 1$, $b = 3$, $c = -5$

$\Delta = 9 + 20 = 29$

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

$\sqrt{29} \approx 5.385$ (keep an extra decimal place until the last step)

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

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

Worked Example 7: No Real Solution

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

$\Delta = 4 - 20 = -16$

Since $\Delta < 0$, there are no real roots. In CAPS, this is a complete and correct answer — you are not expected to work with complex numbers at school level.

The Discriminant as a Topic in Its Own Right

At matric level, examiners frequently ask questions about the discriminant rather than asking you to solve:

• "For which value(s) of $k$ does $kx^2 + 4x + 1 = 0$ have equal roots?" → set $\Delta = 0$ and solve for $k$
• "Show that the roots of $2x^2 - 3x + 5 = 0$ are non-real." → calculate $\Delta$ and show it is negative
• "For what values of $p$ will $px^2 + 6x + 3 = 0$ have two distinct real roots?" → set $\Delta > 0$ and solve the inequality

These questions always start with $\Delta = b^2 - 4ac$. If you've practised this calculation until it's automatic, you'll earn full marks.

The Five Most Common Mistakes

1. Not rearranging to standard form first. If there are terms on the right-hand side, move them before doing anything else.

2. Wrong sign for $-b$ in the formula. If $b = -5$, then $-b = +5$. Rewrite $-(-5)$ explicitly in your working so you can see the sign.

3. Dividing by $2a$ too early. The entire numerator $(-b \pm \sqrt{\Delta})$ must be divided by $2a$. Use brackets: $(-b \pm \sqrt{\Delta}) / (2a)$.

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

5. Rounding at an intermediate step. Keep exact values or at least one extra decimal place throughout, then round only the final answer.

What the Examiner Allocates Marks For

A typical quadratic formula question in CAPS Paper 1 (4–6 marks) is marked roughly as follows:

• Setting up in standard form: 1 mark
• Correct substitution into the formula: 1 mark
• Correct discriminant / simplification: 1 mark
• Both correct final answers: 2 marks

You can earn 3 of 4 marks even if you make an arithmetic error at the end — provided your method is clearly shown. Never leave a quadratic blank.

Practice Problems

Work through these before your next test. Cover your solutions and only check once you have a final answer:

1. $x^2 - 9x + 20 = 0$ (factors neatly — verify with formula)
2. $2x^2 + 5x - 3 = 0$ (formula recommended)
3. $x^2 - 6x + 9 = 0$ (discriminant tells you what to expect)
4. $3x^2 = 5x + 2$ (rearrange first)
5. $x^2 + x + 1 = 0$ (show the roots are non-real)
6. For which value of $k$ does $x^2 - kx + 9 = 0$ have equal roots?

Need More Help?

Quadratic equations feed directly into functions, calculus, and analytical geometry at matric level — the better your foundation here, the easier those topics become. If you want to work through exam-style questions with immediate feedback, book a session. I cover Grade 10–12 mathematics in English and Afrikaans, for both NSC and IEB.

There's also a full guide specifically on using the quadratic formula step by step if you want to go deeper on that method.