Quadratic equations Revision Notes for AQA GCSE Further Maths

Quadratic equations

Introduction to quadratic equations

A quadratic equation is any equation that can be written in the standard form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ . These equations are fundamental in algebra because they appear frequently in real-world problems involving areas, projectile motion, and optimisation.

There are three main methods for solving quadratic equations: factorising, completing the square, and using the quadratic formula. Each method has its advantages depending on the specific equation you're working with.

Method 1: Solving by factorising

Factorising works by using a fundamental property of numbers called the zero product property. This states that if the product of two or more numbers equals zero, then at least one of those numbers must be zero.

When solving by factorisation, you need to rearrange the equation so that all terms are on one side, leaving zero on the other side. Then you factor the expression and set each factor equal to zero.

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Worked Example: Solving $x^2 = 4x + 21$

Step 1: Move all terms to one side $x^2 - 4x - 21 = 0$

Step 2: Find two numbers that multiply to $-21$ and add to $-4$ These numbers are $+3$ and $-7$ : $3 \times (-7) = -21$ and $3 + (-7) = -4$

Step 3: Factor the expression $(x + 3)(x - 7) = 0$

Step 4: Apply the zero product property Either $x + 3 = 0$ or $x - 7 = 0$

Step 5: Solve for $x$ $x = -3$ or $x = 7$

chatImportant

Before attempting to factorise, always ensure all non-zero terms are moved to one side of the equation, leaving only zero on the other side.

Sometimes factorising requires finding two numbers whose sum and product match specific values. For instance, if you need numbers that add to $10$ and multiply to $-24$ , you would systematically check factor pairs of $24$ until you find $12$ and $-2$ .

However, not all quadratic equations can be easily factorised, especially when the coefficients don't work out to nice whole numbers. In these cases, you'll need to use completing the square or the quadratic formula.

Method 2: Completing the square

Completing the square is a powerful algebraic technique that transforms a quadratic equation into a perfect square form. This method is particularly useful when factorising doesn't work easily or when you need to find the exact form of solutions.

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Worked Example: Solving $x^2 - 8x + 3 = 0$

Step 1: Move the constant to the right side $x^2 - 8x = -3$

Step 2: Complete the square Take the coefficient of $x$ (which is $-8$ ), halve it to get $-4$ , then square it: $(-4)^2 = 16$

Step 3: Add this value to both sides $x^2 - 8x + 16 = -3 + 16$ $x^2 - 8x + 16 = 13$

Step 4: Factor the perfect square $(x - 4)^2 = 13$

Step 5: Take the square root of both sides $x - 4 = \pm\sqrt{13}$

Step 6: Solve for $x$ $x = 4 \pm \sqrt{13}$

The process follows a specific pattern. You consider the coefficient of the $x$ term, halve it, then square that result. This method works because you're essentially building a perfect square trinomial.

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If the coefficient of $x^2$ is not $1$ , you must first divide the entire equation by that coefficient before completing the square.

Method 3: Using the quadratic formula

The quadratic formula is the most universal method for solving quadratic equations. It works for any quadratic equation, regardless of whether it can be factorised or easily completed to a square.

The formula is:

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

This formula is derived by completing the square on the general form $ax^2 + bx + c = 0$ , which means it contains all the algebraic work already done for you.

To use the formula, you first identify the values of $a$ , $b$ , and $c$ from your equation when it's in standard form. Then substitute these values directly into the formula.

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Worked Example: Solving $3x^2 - 4x - 2 = 0$

Step 1: Identify the coefficients $a = 3$ , $b = -4$ , $c = -2$

Step 2: Substitute into the quadratic formula $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-2)}}{2(3)}$

Step 3: Simplify $x = \frac{4 \pm \sqrt{16 + 24}}{6}$ $x = \frac{4 \pm \sqrt{40}}{6}$

Therefore: $x = \frac{4 \pm \sqrt{40}}{6}$

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The Discriminant: $b^2 - 4ac$

The expression under the square root tells you important information about the solutions:

If $b^2 - 4ac > 0$ , there are two different real solutions
If $b^2 - 4ac = 0$ , there is one repeated real solution
If $b^2 - 4ac \< 0$ , there are no real solutions

Graphical representation of quadratic functions

Quadratic equations are closely related to quadratic functions of the form $y = ax^2 + bx + c$ . When you graph such a function, you get a U-shaped curve called a parabola.

The solutions to the quadratic equation $ax^2 + bx + c = 0$ correspond to the x-intercepts of the parabola - the points where the curve crosses the x-axis. This is because these are the points where $y = 0$ .

The parabola has several important features:

The vertex is the lowest (or highest) point on the curve, located at $x = -\frac{b}{2a}$
The axis of symmetry is a vertical line through the vertex
The two x-intercepts (if they exist) are equidistant from the axis of symmetry

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Understanding this graphical representation helps you visualise why quadratic equations can have two solutions, one solution, or no real solutions, depending on how many times the parabola crosses the x-axis.

Real-world applications

Quadratic equations frequently appear in practical problems involving areas, dimensions, and optimisation.

For instance, when working with geometric shapes like triangles where you know relationships between the dimensions, you often end up with quadratic equations. If a triangle has a height of $x$ cm and a base of $(2x + 1)$ cm, and you know its area, you can set up an equation using the area formula and solve for $x$ .

Similarly, problems involving rectangular areas, projectile motion, or profit maximisation often lead to quadratic equations. The key is identifying the relationships between the variables and setting up the equation correctly.

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Problem-Solving Steps for Real-World Applications:

Define your variable clearly
Set up the equation based on the given relationships
Solve using the most appropriate method
Check that your answer makes sense in the context (for example, lengths must be positive)

Key takeaways

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Key Points to Remember:

Three main methods exist for solving quadratic equations: factorising (fastest when it works), completing the square (shows the structure clearly), and the quadratic formula (works for all equations)
The zero product property is fundamental to factorising: if two expressions multiply to give zero, at least one must equal zero
The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ works for any quadratic equation when written in the form $ax^2 + bx + c = 0$
The discriminant $(b^2 - 4ac)$ tells you about the nature of solutions: positive means two real solutions, zero means one repeated solution, negative means no real solutions
Quadratic equations represent parabolas graphically, and the solutions are the x-intercepts where the curve crosses the x-axis

Quadratic equations (AQA GCSE Further Maths): Revision Notes