The Quadratic Formula Revision Notes for AQA GCSE Maths

The Quadratic Formula

What is the quadratic formula?

The quadratic formula is a powerful mathematical tool that allows you to find the solutions to any quadratic equation. A quadratic equation is one that can be written in the standard form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are numbers (called coefficients) and $a \neq 0$ .

The quadratic formula itself is:

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

This formula will give you the solutions (also called roots) for any quadratic equation, even when other methods like factorisation don't work easily.

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The quadratic formula is considered one of the most important formulas in mathematics because it provides a universal method for solving quadratic equations. Unlike factorisation, which only works for certain equations, the quadratic formula works for every quadratic equation.

Five crucial details to remember

When using the quadratic formula, there are some important points that can help you avoid common mistakes. These details are essential for getting the right answer every time.

1. Take your time

Always work through the formula step by step, writing down each stage clearly as you go. Rushing through can lead to careless errors, so it's better to be methodical and accurate.

2. Watch out for minus signs

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WHENEVER YOU GET A MINUS SIGN, THE ALARM BELLS SHOULD ALWAYS RING!

This is absolutely critical! Minus signs are the most common source of errors in quadratic formula calculations. Double-check your work whenever they appear, especially when substituting negative values for coefficients.

3. Remember the denominator

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The bottom of the fraction is 2a, not just a. This means you need to divide the entire numerator by $2a$ . Many students forget to include the "2" part, which leads to incorrect answers.

4. The ± symbol gives two solutions

The plus-minus sign (±) in the formula means you'll get two different answers. In your final step, you need to calculate both possibilities: one using the plus sign and one using the minus sign.

5. Check for negative values under the square root

If you get a negative number inside the square root (the discriminant $b^2 - 4ac$ ), go back and check your working. Some quadratic equations do have negative discriminants, but they won't appear in GCSE questions typically.

When should you use the quadratic formula?

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The quadratic formula is particularly useful in certain situations where other methods might not work as well or when specific types of answers are required.

You should use the quadratic formula when:

You have a quadratic equation that doesn't factorise easily
The question asks for answers to a specific number of decimal places
The question asks for answers to a certain number of significant figures
The question asks for exact answers or answers in surd form

The quadratic formula is especially valuable because it works for every quadratic equation, regardless of whether the equation can be factorised or not.

Step-by-step example

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Worked Example: Solving a Quadratic Equation

Problem: Solve $3x^2 + 7x = 1$ , giving your answers to 2 decimal places.

Step 1: Rearrange into standard form First, we need to get the equation into the form $ax^2 + bx + c = 0$ : $3x^2 + 7x - 1 = 0$

Step 2: Identify the coefficients From our equation $3x^2 + 7x - 1 = 0$ , we can identify:

$a = 3$
$b = 7$
$c = -1$

Step 3: Substitute into the formula Now we put these values into the quadratic formula: $x = \frac{-7 \pm \sqrt{7^2 - 4 \times 3 \times (-1)}}{2 \times 3}$

Step 4: Calculate the discriminant Work out what's under the square root first: $7^2 - 4 \times 3 \times (-1) = 49 + 12 = 61$

Step 5: Complete the calculation $x = \frac{-7 \pm \sqrt{61}}{6}$

Since $\sqrt{61} \approx 7.81$ , we get: $x = \frac{-7 + 7.81}{6} = \frac{0.81}{6} = 0.135...$ or $x = \frac{-7 - 7.81}{6} = \frac{-14.81}{6} = -2.468...$

Step 6: Round to the required accuracy To 2 decimal places: $x = 0.14$ or $x = -2.47$

Step 7: Check your answer It's always good practice to substitute your answers back into the original equation to verify they work.

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Key Points to Remember:

The quadratic formula works for ANY quadratic equation in the form $ax^2 + bx + c = 0$
Be especially careful with minus signs - they're the most common source of errors
The denominator is $2a$ , not just $a$
The ± symbol means you'll get two solutions
Use the quadratic formula when factorisation is difficult or when you need decimal/exact answers

The Quadratic Formula (AQA GCSE Maths): Revision Notes