The Quadratic Formula Explained Step by Step

The quadratic formula is one of the most important tools in algebra. It lets you solve any quadratic equation, even the ones that do not factorise neatly. If you are in Year 10 or Year 11, this is a formula you will use again and again, so it is worth understanding properly rather than just memorising.

What is a quadratic equation?

A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are numbers, and a is not zero. The squared term is what makes it quadratic. Examples include:

• x² + 5x + 6 = 0 (here a = 1, b = 5, c = 6)
• 2x² - 3x - 2 = 0 (here a = 2, b = -3, c = -2)
• x² - 7 = 0 (here a = 1, b = 0, c = -7)

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The formula

x = (-b ± √(b² - 4ac)) / 2a

The "±" is critical. It means you get two solutions: one where you add the square root and one where you subtract it. This makes sense because a quadratic equation can have up to two solutions (the parabola can cross the x-axis in two places).

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Worked example 1: A straightforward quadratic

Solve x² + 5x + 6 = 0.

Step 1: Identify a, b, and c.

a = 1, b = 5, c = 6

Step 2: Calculate the discriminant (b² - 4ac).

Discriminant = 5² - 4(1)(6) = 25 - 24 = 1

Step 3: Substitute into the formula.

x = (-5 ± √1) / (2 × 1)

x = (-5 ± 1) / 2

Step 4: Calculate both solutions.

x = (-5 + 1) / 2 = -4 / 2 = -2

x = (-5 - 1) / 2 = -6 / 2 = -3

The solutions are x = -2 and x = -3.

You can check by factorising: x² + 5x + 6 = (x + 2)(x + 3) = 0. Same answers.

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Worked example 2: When factorising does not work easily

Solve 2x² + 3x - 4 = 0.

Step 1: a = 2, b = 3, c = -4

Step 2: Discriminant = 3² - 4(2)(-4) = 9 + 32 = 41

Step 3: x = (-3 ± √41) / (2 × 2)

x = (-3 ± 6.403) / 4

Step 4:

x = (-3 + 6.403) / 4 = 3.403 / 4 = 0.85 (to two decimal places)

x = (-3 - 6.403) / 4 = -9.403 / 4 = -2.35 (to two decimal places)

This equation does not factorise into nice whole numbers, which is exactly when the quadratic formula becomes essential.

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Worked example 3: No real solutions

Solve x² + 2x + 5 = 0.

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

Step 2: Discriminant = 2² - 4(1)(5) = 4 - 20 = -16

The discriminant is negative. You cannot take the square root of a negative number (at least not with real numbers), so this equation has no real solutions. Graphically, this means the parabola does not cross the x-axis at all.

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The discriminant: your preview of the answer

The expression under the square root sign (b² - 4ac) is called the discriminant. It tells you how many solutions to expect before you even finish the calculation:

Discriminant value	Number of solutions	What it means
Positive (greater than 0)	Two distinct solutions	Parabola crosses x-axis twice
Zero	One repeated solution	Parabola touches x-axis once
Negative (less than 0)	No real solutions	Parabola does not reach x-axis

Calculating the discriminant first is a smart habit. It tells you what to expect and helps you catch errors. If you get a negative discriminant but somehow find two solutions, you know something went wrong.

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When to use the quadratic formula

You have several methods for solving quadratics:

1. Factorising: Quickest when it works, but only works for equations with neat integer solutions.
2. Completing the square: Useful for deriving the formula and for certain applications, but more steps.
3. The quadratic formula: Works for every quadratic equation, every time.

In exams, try factorising first. If you cannot spot the factors within about 30 seconds, switch to the formula. It is reliable and systematic, which is exactly what you want under time pressure.

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Common mistakes to avoid

• Getting the sign of b wrong. The formula starts with -b, so if b is already negative, -b becomes positive. For example, if b = -3, then -b = 3. Write it out carefully.
• Forgetting that the denominator is 2a, not just 2. If a = 3, the denominator is 6, not 2. This is a very common slip.
• Not rearranging to standard form first. The formula only works when the equation equals zero. If you have 3x² = 5x + 2, rearrange to 3x² - 5x - 2 = 0 before identifying a, b, and c.
• Dropping the negative in -4ac. When c is negative, -4ac becomes positive. Watch the double negative carefully: -4 × 2 × (-3) = +24, not -24.

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Building confidence with quadratics

The quadratic formula rewards careful, methodical working. Write out a, b, and c clearly. Calculate the discriminant as a separate step. Substitute carefully into the formula. Check your answer by plugging it back into the original equation.

If your child is studying algebra in Year 10 or 11 and wants structured practice with quadratics, imSteyn guides students through quadratic problems step by step. Rather than showing the solution, it helps students identify a, b, and c themselves and work through the formula with support, building the kind of fluency that leads to exam confidence.