How to Solve: Quadratic Functions (Discriminant, Completing the Square, Simultaneous) – GCSE/A-Level Maths Guide

How to Solve: Quadratic Functions (Discriminant, Completing the Square, Simultaneous) – GCSE/A-Level Maths Guide

Introduction

Mastering quadratics unlocks real-world problems—from calculating projectile motion (like a basketball shot) to optimising profit in business. In exams, quadratics appear in 10-15% of GCSE/A-Level Maths questions, often worth 8-12 marks—so losing marks here means losing grades. Today, you’ll learn three core methods to solve any quadratic problem with confidence.

What You Need To Know First

Before diving in, ensure you understand: 1. Basic algebra (expanding brackets, solving linear equations). 2. Quadratic form: What ( ax^2 + bx + c = 0 ) looks like and how to identify ( a, b, c ). 3. Graphs of quadratics: The shape (parabola), vertex, and roots.

Key Vocabulary

Formulas To Know

1. Quadratic Formula

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ] - ( a, b, c ): Coefficients from ( ax^2 + bx + c = 0 ). - MEMORISE THIS (given on some exam sheets, but know it by heart).

2. Discriminant

[ D = b^2 - 4ac ] - MEMORISE THIS. - Tells you the number of roots: - ( D > 0 ): Two distinct real roots. - ( D = 0 ): One real root (repeated). - ( D < 0 ): No real roots (complex roots).

3. Completing the Square (General Form)

[ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) ] - Given on exam sheet (but practice deriving it).

4. Simultaneous Equations (Substitution)

• Solve one equation for ( y ) (or ( x )), then substitute into the other.

Step-by-Step Method

Method 1: Using the Discriminant

When to use: To find the number of roots or nature of roots without solving the equation.

Steps: 1. Write the quadratic in standard form: ( ax^2 + bx + c = 0 ). 2. Identify ( a, b, c ). 3. Calculate the discriminant: ( D = b^2 - 4ac ). 4. Interpret ( D ):
- ( D > 0 ): Two distinct real roots.
- ( D = 0 ): One real root (repeated).
- ( D < 0 ): No real roots.

Worked Example: Find the number of roots of ( 2x^2 - 4x + 3 = 0 ). 1. ( a = 2 ), ( b = -4 ), ( c = 3 ). 2. ( D = (-4)^2 - 4(2)(3) = 16 - 24 = -8 ). 3. ( D < 0 ), so no real roots.

Method 2: Completing the Square

When to use: To find the vertex of a parabola or solve quadratics when factoring is hard.

Steps: 1. Write the quadratic in the form ( ax^2 + bx + c ). 2. If ( a \neq 1 ), factor ( a ) out of the first two terms. 3. Take half of ( b ), square it, and add/subtract inside the bracket. 4. Rewrite as ( a(x + p)^2 + q ). 5. Solve for ( x ) if needed (set ( = 0 )).

Worked Example: Solve ( x^2 + 6x + 5 = 0 ) by completing the square. 1. ( x^2 + 6x + 5 ). 2. Half of ( 6 ) is ( 3 ), square it: ( 9 ). 3. ( x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4 ). 4. Set ( = 0 ): ( (x + 3)^2 - 4 = 0 ). 5. ( (x + 3)^2 = 4 ). 6. ( x + 3 = \pm 2 ). 7. ( x = -3 \pm 2 ), so ( x = -1 ) or ( x = -5 ).

Method 3: Solving Simultaneous Equations (Quadratic + Linear)

When to use: When you have one quadratic and one linear equation (e.g., ( y = x^2 ) and ( y = 2x + 3 )).

Steps: 1. Write both equations clearly. 2. Substitute the linear equation into the quadratic (replace ( y )). 3. Rearrange into standard quadratic form ( ax^2 + bx + c = 0 ). 4. Solve using factoring, quadratic formula, or completing the square. 5. Find ( y ) by substituting ( x ) back into the linear equation.

Worked Example: Solve ( y = x^2 ) and ( y = 4x - 3 ) simultaneously. 1. Substitute ( y = 4x - 3 ) into ( y = x^2 ): ( 4x - 3 = x^2 ). 2. Rearrange: ( x^2 - 4x + 3 = 0 ). 3. Factor: ( (x - 1)(x - 3) = 0 ). 4. ( x = 1 ) or ( x = 3 ). 5. Find ( y ):
- If ( x = 1 ), ( y = 4(1) - 3 = 1 ).
- If ( x = 3 ), ( y = 4(3) - 3 = 9 ). 6. Solutions: ( (1, 1) ) and ( (3, 9) ).

Worked Examples

Example 1 – Basic (Discriminant)

Question: Find the number of real roots of ( x^2 - 5x + 6 = 0 ). Solution: 1. ( a = 1 ), ( b = -5 ), ( c = 6 ). 2. ( D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ). 3. ( D > 0 ), so two distinct real roots.

What we did and why: We used the discriminant to quickly check the number of roots without solving the equation.

Example 2 – Medium (Completing the Square)

Question: Rewrite ( 2x^2 + 8x + 3 ) in completed square form. Solution: 1. Factor ( 2 ) from first two terms: ( 2(x^2 + 4x) + 3 ). 2. Half of ( 4 ) is ( 2 ), square it: ( 4 ). 3. Add and subtract ( 4 ) inside the bracket: ( 2(x^2 + 4x + 4 - 4) + 3 ). 4. Rewrite: ( 2((x + 2)^2 - 4) + 3 = 2(x + 2)^2 - 8 + 3 ). 5. Simplify: ( 2(x + 2)^2 - 5 ).

What we did and why: We completed the square to find the vertex form, which helps identify the minimum/maximum point of the parabola.

Example 3 – Exam-Style (Simultaneous Equations)

Question: Find the points of intersection of ( y = x^2 - 2x ) and ( y = x + 4 ). Solution: 1. Set ( x^2 - 2x = x + 4 ). 2. Rearrange: ( x^2 - 3x - 4 = 0 ). 3. Factor: ( (x - 4)(x + 1) = 0 ). 4. ( x = 4 ) or ( x = -1 ). 5. Find ( y ):
- If ( x = 4 ), ( y = 4 + 4 = 8 ).
- If ( x = -1 ), ( y = -1 + 4 = 3 ). 6. Points of intersection: ( (4, 8) ) and ( (-1, 3) ).

What we did and why: We solved a quadratic and linear equation simultaneously to find where two graphs intersect.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute checklist for quadratics: 1. Discriminant: ( D = b^2 - 4ac ). If ( D > 0 ), two roots; ( D = 0 ), one root; ( D < 0 ), no real roots. 2. Quadratic formula: ( x = \frac{-b \pm \sqrt{D}}{2a} ). Memorise it—it works every time. 3. Completing the square: Halve ( b ), square it, add/subtract. Useful for finding the vertex. 4. Simultaneous equations: Substitute the linear into the quadratic, solve, then find ( y ). 5. Common mistakes: Forgetting the ( \pm ), sign errors, and not factoring out ( a ). Double-check your work! You’ve got this—go smash that exam!