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
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.