🎯 Learning Objectives
- Write a quadratic in standard form.
- Substitute coefficients into the quadratic formula.
- Interpret two, one or no real solutions.
Introduction
This lesson develops The Quadratic Formula from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
🤔 Start with an Idea
Factorisation is elegant, but not every quadratic factorises neatly. The quadratic formula gives a reliable method for every equation in the form ax² + bx + c = 0.
📖 Key Notes
x = (−b ± √(b² − 4ac)) / 2a
The expression b² − 4ac is the discriminant. It tells us how many real roots are possible.
🧮 Formula Stepper
x² + 5x + 2 = 0
First identify a, b and c.
🧠 Let's Think Together
Solve: 2x² − 3x − 2 = 0.
- a = 2, b = −3, c = −2.
- Discriminant = (−3)² − 4(2)(−2) = 25.
- x = [3 ± 5]/4.
x = 2 or x = −1/2
✍️ Worked Examples
3x² + x − 1 = 0 gives x = (−1 ± √13)/6.
🎯 Practice Questions
Easy
Solve x² + 3x − 4 = 0 using the formula.
Show answer
x = 1 or x = −4
Medium
Solve 2x² + x − 3 = 0.
Show answer
x = 1 or x = −3/2
⚠️ Common Mistakes
- Substituting b without its sign.
- Forgetting that the whole numerator is divided by 2a.
- Using ± for only the square-root term incorrectly.
📝 Exam Focus
Write a, b and c on a separate line before substitution. Give exact surd answers unless a decimal approximation is requested.
Summary
- Put the equation equal to zero.
- Identify signed coefficients.
- Calculate the discriminant carefully.
- Use both ± branches.