Three Ways to Solve a Quadratic Equation

A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \ne 0$. There are three standard ways to solve one: factoring, the quadratic formula and completing the square. Factoring is fastest when it works, the formula always works, and completing the square explains where the formula comes from.

Before you start: get it into standard form

Move everything to one side so the other side is zero. For example, $x^2 = 5x - 6$ becomes

Now you can read off $a = 1$, $b = -5$ and $c = 6$.

Method 1: Factoring

Look for two numbers that multiply to $c$ and add to $b$.

For $x^2 - 5x + 6 = 0$ we need two numbers that multiply to $6$ and add to $-5$. Those are $-2$ and $-3$:

If a product is zero, one of the factors must be zero, so $x = 2$ or $x = 3$.

Second example: $x^2 + x - 6 = 0$. Two numbers that multiply to $-6$ and add to $1$ are $3$ and $-2$:

Method 2: The quadratic formula

This works for every quadratic:

Example: $x^2 - 2x - 1 = 0$, so $a = 1$, $b = -2$, $c = -1$.

These roots are irrational, which is why factoring would not have found them. As decimals they are about $2.414$ and $-0.414$.

Method 3: Completing the square

Rewrite the quadratic as a perfect square plus a constant.

For $x^2 - 2x - 1 = 0$:

1. Move the constant: $x^2 - 2x = 1$.
2. Add the square of half the $x$-coefficient to both sides. Half of $-2$ is $-1$, and $(-1)^2 = 1$: $x^2 - 2x + 1 = 2$.
3. The left side is now a square: $(x - 1)^2 = 2$.
4. Take square roots: $x - 1 = \pm\sqrt{2}$, so $x = 1 \pm \sqrt{2}$.

Same answer as the formula, because the formula is completing the square done once in general.

Which method should I use?

Common mistakes

• Forgetting the ± sign. A quadratic usually has two roots.
• Sign errors with $-b$. If $b = -5$, then $-b = 5$.
• Dividing only part of the numerator by $2a$. The whole of $-b \pm \sqrt{\ldots}$ is divided.

Practice questions

1. Solve $x^2 - 7x + 12 = 0$ by factoring.
2. Solve $2x^2 + 3x - 2 = 0$ with the formula.
3. Solve $x^2 + 6x + 4 = 0$ by completing the square.

Answers: 1) $x = 3$ or $x = 4$ 2) $x = -2$ or $x = \frac{1}{2}$ 3) $x = -3 \pm \sqrt{5}$

You can check any of these with the quadratic equation solver, which shows the parabola and its roots, or plot the curve in the graphing calculator. For more on the $b^2 - 4ac$ part, read what the discriminant tells you.

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