2.4 Solving Quadratic Equations

---

📘 Main Content

---

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form:

$ax^2 + bx + c = 0$

where:

• $a$, $b$, and $c$ are constants (fixed numbers)
• $a \neq 0$ (if $a$ were zero, there would be no $x^2$ term and it wouldn't be quadratic)
• $x$ is the unknown (the value you are trying to find)

Examples of quadratic equations:

• $x^2 - 5x + 6 = 0$
• $2x^2 + 3x - 2 = 0$
• $x^2 - 9 = 0$

The solutions of a quadratic equation are called its roots. A quadratic equation can have two roots, one repeated root, or no real roots at all.

---

How Factorisation Works — The Key Idea

If you can write a quadratic as:

$(x - p)(x - q) = 0$

then either $(x - p) = 0$ or $(x - q) = 0$.

So the roots are $x = p$ or $x = q$.

This is called the Zero Product Property — if two things multiplied together equal zero, then at least one of them must be zero.

---

How to Factorise $x^2 + bx + c = 0$ (when $a = 1$)

You need to find two numbers that:

• Multiply to give $c$ (the constant term)
• Add to give $b$ (the coefficient of $x$)

---

Example 1: Solve $x^2 - 5x + 6 = 0$

Step 1: Find two numbers that multiply to $+6$ and add to $-5$. Those numbers are $-2$ and $-3$, because $(-2) \times (-3) = 6$ and $(-2) + (-3) = -5$.

Step 2: Write in factorised form: $(x - 2)(x - 3) = 0$

Step 3: Set each bracket equal to zero: $x - 2 = 0 \quad \Rightarrow \quad x = 2$ $x - 3 = 0 \quad \Rightarrow \quad x = 3$

✅ Roots: $x = 2$ or $x = 3$