Have you ever heard of a quadratic equation and thought, “That sounds complicated”? Don’t worry—you’re not alone. But here’s the secret: quadratic equations are like puzzles.

In this blog, we’ll break down quadratic equations into simple steps. We’ll start by understanding what they are, then look at ways to factorise and solve them. Along the way, you’ll discover how these equations appear in real life. For example, they are used when predicting the shape of a rollercoaster or the flight of a ball. So grab your thinking cap, and let’s dive into the world of quadratics together!

FAQs

What’s the easiest way to solve a quadratic equation?

Factorisation is the quickest, but it doesn’t work for all equations.

Why do some quadratics have no solutions?

If the graph never touches the x-axis, there are no real solutions.

How can I check if my answers are correct?

Substitute your answers back into the original equation.

How can I check if my answers are correct?

Substitute your answers back into the original equation.

Do I need to memorise the quadratic formula?

Yes, it’s essential for exams. Practicing it will make it easier.

Can I use a calculator for all quadratic equations?

Calculators can solve them, but understanding the methods is crucial for exams.

What Are Quadratic Equations?

Imagine you’re drawing a curve on a graph. Sometimes, that curve looks like a happy smile or a sad frown. That’s the shape you get when you graph a quadratic equation.

A quadratic equation is just a fancy way of describing an equation with an x² in it. For example: x²+3x+2=0

Mastering the Art of Rearranging Equations: A Guide for GCSE Maths Students

Why Are Quadratic Equations Important?

Quadratic equations are everywhere! They help us understand how things fly (like rockets or basketballs), calculate areas, and even predict profits for businesses.

Learning to solve these equations is like unlocking a secret code to understanding how the world works.

Breaking Down Quadratic Equations

The standard form of a quadratic equation is: ax²+bx+c=0

Here’s what the letters mean:

• a: The number in front of x². It tells us how “steep” the curve is.
• b: The number in front of x.
• c: The constant (a plain number).

How Do We Solve Quadratic Equations?

There are a few ways to solve quadratic equations. Don’t worry—they’re easier than they sound!

1. Factorising (Splitting into Brackets)

Think of factorising as a puzzle. You break the quadratic equation into two smaller brackets.

Example: x²+5x+6=0

Ask yourself:

• What two numbers multiply to 6 but add to 5?
  Answer: 2 and 3!

So, we write: (x+2)(x+3)=0

To solve it, set each bracket to zero:

x+2=0  ⟹ (Rearrange for x)  x=−2

x+3=0  ⟹  (Rearrange for x) x=−3

The answers are x=−2 and x=−3.

2. Using the Quadratic Formula

Sometimes, factorising doesn’t work. That’s when we use a formula:

x = (-b ± √(b² – 4ac)) / (2a)

It looks tricky, but don’t worry—your calculator can help!

Let’s break the calculation down for: x²+7x+10=0 using the quadratic formula into super simple steps.

STEP 1: Start with the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)

STEP 2: Identify the values of a, b, and c (from the standard form mentioned earlier):
a=1, b=7, c=10

Substitute these into the formula:
x = −7±√7² −4(1)(10) / 2(1)

Simplify inside the square root:
7² =49
-4(1)(10) = -40
√49-40 = √9 = 3

Write the simplified equation:
x=−7±3 / 2

Solve for both possible values of x:
x = −7+3 / 2 = -4/2 = -2
x = −7−3 / 2= −10/2 = −5

Final Answer:
x=−2 and x=−5

3. Completing the Square

This method rewrites the equation into a neat “perfect square.”

Example: x²+6x+5=0

Rearrange: (x+3)²−4=0

Solve: (x+3)²=4

x+3=±2  ⟹ (Rearranging for x):  x=−3+2=−1 or x=−3−2=−5

In-depth steps for completing the square

You started with the quadratic equation: x²+6x+5=0

Step 1: Rearranging the terms

To complete the square, you need to focus on the x² and x terms. First, move the constant to the other side: x²+6x=−5

Step 2: Completing the square

Now, you want to make the left side a perfect square. To do this, take half of the coefficient of x (which is 6), square it, and add that number to both sides:

Half of 6 is 3, and 3²=9.

So, add 9 to both sides of the equation: x²+6x+9=−5+9

This simplifies to: (x+3)²=4

Step 3: Solving for x

Now, take the square root of both sides: x+3=±2. Note how when we take the square root, the power disappears. They cancel out on the left-hand side of the equation.

Step 4: Solving the equation

Now, solve for x by isolating it:

• x+3=2  ⟹  x=−1
• x+3=−2  ⟹  x=−5

Final Answer:

x=−1 or x=−5

The two possible values for x are indeed -1 and −5.

Special Quadratics: Difference of Two Squares

Some quadratics are extra easy to solve, like x²−9.

Here’s a trick: a²−b² = (a+b)(a−b)

At first glance, this looks a bit tricky. However, it’s actually super easy to solve. This is thanks to a special trick called the difference of squares. Here’s the secret:

The Trick: a² – b² = (a + b)(a – b)

This is a special rule in maths that says:

• If you have a² (a number squared) and b² (another number squared), you can factor the difference. It can be broken down into smaller parts: (a + b)(a – b).

Now, let’s use this trick on your example:

Step 1: Recognising the Squares

Look at x² – 9:

• x² is already a perfect square (it’s x times x).
• 9 is also a perfect square because 3 × 3 = 9.

So, we can rewrite this equation as:

x² – 3²

Step 2: Applying the Formula

Now that we have a² – b², we can use our trick!

We can think of a as x and b as 3. So, we rewrite the equation like this:

(x + 3)(x – 3)

Step 3: Finding the Solutions

Now, we have (x + 3)(x – 3) = 0. This is the key part: if two things multiplied together give you 0, one of them must be 0.

So, we have two possible situations:

1. x + 3 = 0, which gives x = -3
2. x – 3 = 0, which gives x = 3

Final Answer:

The solutions to x² – 9 = 0 are x = -3 and x = 3.

Why Does This Work?

This trick works because of how multiplication and factoring work together. When you multiply (x + 3) and (x – 3), you get:

x² – 3x + 3x – 9, which simplifies to x² – 9 — exactly what we started with! That’s why we can split it up like this.

Recap of the Steps:

1. Recognise a² – b² (in this case x² – 9).
2. Use the formula a² – b² = (a + b)(a – b).
3. Set each part equal to 0 and solve for x.

So, x = -3 and x = 3 are the two solutions. Plug this back into the initial formula to check your answer equals 0.
Remember:
x² -9
Plugging in x=3 –>3²-9 = 9-9 = 0
Plugging in x=-3 –> (-3)²-9 = 9-9=0.
Remember that squaring a negative number produces a positive result.

Why is this “extra easy”?

This is considered an easy type of quadratic. We don’t need to use the usual long methods like factoring by grouping or completing the square. We just use the difference of squares trick!

Why Are Quadratic Equations Useful in Real Life?

Quadratic equations help us:

1. Plan Motion: Where will a football land?
2. Maximise Profits: How can a business make the most money?
3. Design Buildings: Engineers use quadratics to calculate structure shapes.

Common Mistakes to Avoid

1. Forgetting to Set the Equation Equal to Zero:
   Always start with ax²+bx+c=0.
2. Mixing Up Signs:
   Double-check your factor pairs. For example, -3 and -2 multiply to 6 but add to -5, not 5.
3. Ignoring Negative Roots:
   Remember, quadratic equations often have two solutions.

How Apollo Scholars Can Help

At Apollo Scholars, we make maths fun and easy to understand. We provide personalised lessons, study support and academic tutoring to help you conquer quadratic equations. Whether you’re a Key Stage 3 or GCSE student, we’re here to guide you every step of the way.