In this section, you can use the practice tool to improve your quadratic equation-solving skills. A quadratic equation has the form

$$ \begin{align} ax^2 + bx + c &= 0 \; \; \color{#E90F89}{\text{ or}} \\[5pt] ax^2 + bx &= c, \end{align}$$

where a, b and c are parameters.

Sometimes these equations are factorable, that is reducible to a product of binomials like $(x + d)(x + e) = 0,$ so that they're fairly easy to solve. But most of the time they are not, so you'll need to learn to solve them by either completing the square (covered here) or using the Po Shen Loh method.

Completing the square

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The basic idea behind completing the square is that if we take a look at a perfect square of binomials, like

$$(x + m)(x + m) = x^2 + 2m(x) + m^2,$$

(m is a constant) we notice that the constant term, $m^2,$ is exactly the square of half of what multiplies $x,$ which is $2m.$ That is,

$$\left( \frac{2m}{2} \right)^2 = m^2$$

When we complete the square, we'll force the left side of the equation to be a perfect square, at the expense of collecting some extra constants on the right. That's OK, we can deal with those at the end.

Steps – an example

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Let's do this by example. To solve a quadratic equation like

$$3x^2 - 2x + 7 = 0,$$

which is not factorable (as far as I know – maybe you're better at it than I am), here are the steps:

1. Move the constant to the right side.

$$3x^2 - 2x = -7$$

2. Divide by the coefficient of $x^2.$

$$x^2 - \frac{2}{3} x = -\frac{7}{3}$$

3. Add the square of ½ of the coefficient of x to both sides of the equation.

Notice that half of $\frac{2}{3}$ is $\frac{1}{3},$ a simple thing often missed. Don't forget, as always in algebra, to add it to both sides.

$$x^2 \color{#E90F89}{-} \frac{2}{3} x + \left( \color{#E90F89}{\frac{1}{3}} \right)^2 = -\frac{7}{3} + \frac{1}{9}$$