Quadratic formula - decimal solutions
You’ll often see the quadratic formula when trying to solve equations where something is squared – such as in physics when calculating the path of a thrown object, or in engineering when designing curves. It’s a powerful tool that always works, even when factorising doesn’t! Jump to the questions
Practise now
Worksheet preview and key skills
Worksheet preview
Practise using the quadratic formula to find decimal solutions with this self-marking maths worksheet.
The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.
What you’ll practise
- Identifying a, b and c.
- Substituting into the quadratic formula.
- Calculating both solutions.
- Rounding decimal solutions appropriately.
Use the interactive worksheet below, or read the Topic guide for the method and worked example.
Solve each equation and round both solutions to 1 decimal place.
Topic guide
What this worksheet practises
This worksheet focuses on using the Quadratic Formula to solve complex quadratic equations (ax² + bx + c = 0) that cannot be factorised. This is a calculator-heavy topic where precision is absolutely vital.
Key method
You must substitute your values carefully into the formula: x = (−b ± √(b² − 4ac)) / 2a
- Identify the values for a (the number attached to x²), b (the number attached to x), and c (the number on its own). Ensure the equation equals zero first.
- Calculate the "discriminant" first: b² − 4ac. This is the part that lives inside the square root. Calculating this first drastically reduces calculator errors.
- Substitute everything into the main formula.
- Because of the ± symbol, you must do the final calculation twice: once using a plus sign, and once using a minus sign. This gives you two distinct answers.
Worked example
Solve 2x² + 5x − 4 = 0 to 2 decimal places.
Step 1: Identify a, b, and c.
a = 2, b = 5, c = −4.
Step 2: Calculate the discriminant (b² − 4ac).
(5)² − (4 × 2 × −4)
25 − (−32)
25 + 32 = 57.
Step 3: Put it all into the main formula.
x = (−5 ± √57) / (2 × 2)
x = (−5 ± √57) / 4
Step 4: Calculate the two answers using your calculator.
Plus version: x = (−5 + √57) / 4 = 0.6374...
Minus version: x = (−5 − √57) / 4 = −3.1374...
Step 5: Round to 2 decimal places.
x = 0.64 or x = −3.14.
Common mistakes to avoid
The single most dangerous error involves negative numbers in the discriminant, specifically when squaring a negative 'b'. If b = −3, typing -3² into a calculator gives −9. You must type (−3)² to get the correct answer of +9. A squared number in the formula must always be positive.
How to check your answer
You can verify your answers by substituting them back into the original equation. For example, 2(0.64)² + 5(0.64) − 4. This calculation will equal 0.0192. Because it is extremely close to zero, it proves our rounded answer of 0.64 is correct.