Quadratic Equations Quizzes

Introduction

Quadratic equations are a fundamental part of GCSE Maths, forming the basis for solving a wide range of algebraic problems. Mastering quadratic equations is essential for understanding algebraic manipulation, graph interpretation, and real-world problem solving. They appear in exam questions in both foundation and higher-tier papers, and are crucial for progressing to more advanced topics such as functions, inequalities, and calculus in A-level Maths.

Core Concepts

What is a Quadratic Equation?

A quadratic equation is an equation in which the highest power of the variable is 2. It can be written in the standard form:

$$ ax^2 + bx + c = 0 $$

where:

• a is the coefficient of \(x^2\) (cannot be 0)
• b is the coefficient of \(x\)
• c is the constant term

Example: \(2x^2 + 3x - 5 = 0\)

Key Terminology

• Roots / Solutions: The values of \(x\) that satisfy the equation.
• Discriminant: The expression \(b^2 - 4ac\) determines the nature of the roots.
• Factorisation: Writing the quadratic as a product of two binomials.
• Completing the Square: Rewriting the quadratic in the form \((x + p)^2 = q\).

Rules & Steps

1. Factorisation Method

Factorisation is often the quickest method if the quadratic can be expressed as the product of two binomials.

1. Write the quadratic in standard form: \(ax^2 + bx + c = 0\).
2. Find two numbers that multiply to \(a \cdot c\) and add to \(b\).
3. Split the middle term using these numbers and factorise by grouping.
4. Set each factor equal to 0 and solve for \(x\).

Example:

$$ x^2 + 5x + 6 = 0 $$

• Find two numbers that multiply to 6 and add to 5: 2 and 3.
• Factor: \(x^2 + 2x + 3x + 6 = (x^2 + 2x) + (3x + 6)\)
• Factor each group: \(x(x + 2) + 3(x + 2) = (x + 2)(x + 3)\)
• Set factors to 0: \(x + 2 = 0 \Rightarrow x = -2\), \(x + 3 = 0 \Rightarrow x = -3\)

2. Using the Quadratic Formula

If factorisation is difficult or impossible, the quadratic formula can solve any quadratic:

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

Steps:

1. Identify \(a\), \(b\), and \(c\) from \(ax^2 + bx + c = 0\).
2. Calculate the discriminant: \(D = b^2 - 4ac\).
3. Compute the roots using the formula.

Example:

$$ 2x^2 + 3x - 2 = 0 $$

• \(a = 2\), \(b = 3\), \(c = -2\)
• \(D = 3^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25\)
• \(x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}\)
• \(x = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5\)
• \(x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\)

3. Completing the Square

This method rewrites the quadratic in the form \((x + p)^2 = q\), which can be solved using square roots.

1. Ensure the coefficient of \(x^2\) is 1 (divide through if necessary).
2. Move the constant term to the other side: \(x^2 + bx = -c\).
3. Add \((b/2)^2\) to both sides to complete the square.
4. Factor the left-hand side: \((x + b/2)^2 = \text{new constant}\).
5. Take square roots and solve for \(x\).

Example:

$$ x^2 + 6x + 5 = 0 $$

• Move constant: \(x^2 + 6x = -5\)
• Add \((6/2)^2 = 9\): \(x^2 + 6x + 9 = -5 + 9 = 4\)
• Factor: \((x + 3)^2 = 4\)
• Take square roots: \(x + 3 = \pm 2\)
• Solutions: \(x = -1, -5\)

Worked Examples

1. Factorise and solve: \(x^2 - x - 6 = 0\) $$x^2 - x - 6 = (x - 3)(x + 2) = 0 \Rightarrow x = 3, -2$$
2. Use quadratic formula: \(3x^2 + 2x - 1 = 0\) $$D = 2^2 - 4 \cdot 3 \cdot (-1) = 4 + 12 = 16$$ $$x = \frac{-2 \pm \sqrt{16}}{6} = \frac{-2 \pm 4}{6} \Rightarrow x = \frac{1}{3}, -1$$
3. Complete the square: \(x^2 + 4x - 5 = 0\) $$x^2 + 4x = 5$$ $$x^2 + 4x + 4 = 9$$ $$(x + 2)^2 = 9 \Rightarrow x + 2 = \pm 3 \Rightarrow x = 1, -5$$
4. Real-life application: A ball is thrown and its height in meters is given by \(h = -5t^2 + 20t + 1\). Find when the ball hits the ground. $$-5t^2 + 20t + 1 = 0$$ $$t = \frac{-20 \pm \sqrt{20^2 - 4(-5)(1)}}{2(-5)} = \frac{-20 \pm \sqrt{400 + 20}}{-10}$$ $$t = \frac{-20 \pm \sqrt{420}}{-10} \approx 0.05 \text{ s or } 4.05 \text{ s}$$
5. Higher-tier example: Solve \(2x^2 - 7x + 3 = 0\) using the quadratic formula. $$D = (-7)^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25$$ $$x = \frac{7 \pm 5}{4} \Rightarrow x = 3, \frac{1}{2}$$

Common Mistakes

• Incorrect factor pairs when factorising. Always check multiplication and addition.
• For the quadratic formula, forgetting the negative sign before \(b\).
• Miscomputing the discriminant or failing to simplify square roots.
• Errors in completing the square: not halving the coefficient of \(x\) correctly.
• Mixing up the order of operations when rearranging terms.

Applications

Quadratic equations appear in various real-life situations:

• Projectile motion and physics problems.
• Optimisation problems, e.g., maximizing area or profit.
• Modelling parabolic curves in architecture and engineering.
• Economics: profit functions often take a quadratic form.

Strategies & Tips

• Always simplify the equation to standard form before solving.
• Check if factorisation is possible before using the quadratic formula.
• Use the discriminant to quickly assess the nature of roots:
  • \(D > 0\): two real roots
  • \(D = 0\): one real repeated root
  • \(D < 0\): two complex roots (higher-tier)
• Draw graphs for visualization; the x-intercepts correspond to solutions.
• Practice different methods to gain flexibility in exams.

Summary

Quadratic equations are a vital component of GCSE Maths. Understanding the different methods—factorisation, completing the square, and the quadratic formula—ensures that you can solve any quadratic problem efficiently. Regular practice, attention to common mistakes, and applying these techniques to real-world scenarios will boost your confidence and exam performance. Start by attempting the quizzes and exercises to solidify your understanding and develop fluency in solving quadratic equations.