Solving quadratic equations: three methods

A quadratic equation is one where the highest power of $x$ is $x^2$.

Every quadratic can be rearranged into the standard form:

$ax^2 + bx + c = 0$

Example: in $2x^2 + 5x - 3 = 0$, we have $a = 2$, $b = 5$ and $c = -3$.

Notice that $c$ keeps its minus sign. Signs are where most mistakes happen.

Solving means finding every value of $x$ that makes the equation true. These values are the roots.

• A quadratic usually has two roots.
• Sometimes it has one, and sometimes none.
• On a graph, the roots are where $y = ax^2 + bx + c$ crosses the $x$-axis. That's because $y = 0$ on the $x$-axis.

Before anything else, rearrange so that one side is 0.

Why zero? Because of this fact:

If two numbers multiply to give 0, at least one of them must be 0.

Zero is special here.

If $A \times B = 6$, then $A$ could be almost anything ($1$, $2$, $0.5$, …).

But if $A \times B = 0$, then $A = 0$ or $B = 0$. Nothing else works.

So if we can write the quadratic as (bracket) $\times$ (bracket) $= 0$, it splits into two easy equations.

Writing an expression as brackets multiplied together is called factorising.

Here's when to use each method:

Start by expanding (multiplying out) two brackets, and watch what happens:

$(x + 2)(x + 5) = x^2 + 5x + 2x + 10$

$= x^2 + 7x + 10$

Look at the two numbers in the brackets, $2$ and $5$:

• they add to give the middle number: $2 + 5 = 7$
• they multiply to give the last number: $2 \times 5 = 10$

This always happens when the $x^2$ has no number in front of it ($a = 1$).

Factorising is the same thing run backwards. To factorise $x^2 + bx + c$:

1. Find two numbers that multiply to $c$ and add to $b$.
2. Put them in the brackets: $(x + \text{first})(x + \text{second})$.

With a number in front of $x^2$, like $2x^2 + 5x - 3$, the "add to $b$, multiply to $c$" trick no longer works on its own.

The fix is called splitting the middle term:

1. Multiply $a \times c$.
2. Find two numbers that multiply to $ac$ and add to $b$.
3. Use them to split $bx$ into two terms.
4. Factorise the first two terms, then the last two. A matching bracket appears.
5. Take out that matching bracket.

Why does a matching bracket appear? The two numbers are chosen so that the second pair of terms is the first pair scaled down. That's why the same bracket appears in both. You'll see it happen in the example.