# Completing the Square Practice Math Problems and Examples

Here you can find practice questions for the method of **solving quadratic equations** by **completing the square**. This is one of the most important problems in high school math! Go through the examples and problems we offer, and become a master of completing the square.

## Solving general quadratic equations by completing the square

In high school algebra, you're bound to encounter the problem of **solving general quadratic equations by the "completing the square" method**. A general quadratic equation is an equation involving a quadratic polynomial (so a polynomial of degree two):

where $a$, $b$, $c$ and $d$ are **real coefficients**. It's clear that such an equation can be simplified to get $a=1$ as the leading coefficient and to get the constant term at the right side of the equation equal to $0$. To do that, let's subtract $d$ from both sides and then divide by $a$:

Consequently, it suffices to know how to solve a quadratic equation of the form:

To do it, add $-c+\frac{b^2}{4}$ to both sides:

We can see a perfect square trinomial on the left side:

Thus,

And so now, if the right side is non-negative, we **take the** **square root** **of both sides** — the left side of the equation becoming $|x + \frac{b}{2}|$ — and perform standard calculations to solve the equation. If the right side is negative, the equation has **no real solution** (although it does have **complex solutions**).

## Completing the square practice problems quiz

What term do you have to add to the following expression in order to get a perfect square trinomial?

- $x^2 + 2x$
- $x^2 - 6x$
- $x^2 + 3x$
- $x^2 + 6x +6$
- $x^2 - 2x + 3$

Answers:

- Add $1$ to get the trinomial $(x+1)^2$.
- Add $9$, trinomial: $(x-3)^2$.
- Add $\frac{9}{4}$, trinomial: $(x+\frac{3}{2})^2$.
- Add $3$, trinomial: $(x+3)^2$.
- Subtract $2$, trinomial: $(x-1)^2$.

## Example questions

Here we **solve quadratic equations** by completing the square so that you can learn this method with some examples.

**Example 1.** $x^2 - x + 0.25 = 1$

We can expand the left-hand side as

We can apply the square root to both sides to get $|x - 0.5|=1$ and therefore we know $x - 0.5 = \pm 1$. Therefore, $x = 1.5$ or $x = -0.5$.

**Example 2**. $2x^2 + 4x + 8 = 0$

Let's divide both sides by $2$:

$x^2 + 2x = 0$

Now, we examine the expression $x^2 + 2x$. To produce these terms by the short multiplication formula, we can use $(x + 1)^2 = x^2 + 2x + 1$. As you can see, the constant term is $1$ while in the equation it is $0$ — we need to **complete the square** by adding $1$:

**Example 3.** $x^2 - 8x + 20 = 0$

At the left side we have $x^2 - 8x$, which can be interpreted as a part of the perfect square trinomial $x^2 - 8x + 16 = (x-4)^2$. However, we have $20$ in our equation and only $16$ in the perfect square trinomial. So let's subtract $4$ from both sides:

Clearly, this equation has no solution in real numbers, because no real number can give $-4$ when squared. Let's solve this equation in complex numbers (if you're not yet familiar with complex numbers, feel free to skip this part). So, we have

## Practice questions

Solve using the method of completing the square:

- $x^2 + 8x - 9 = 0$
- $3x^2 + 12x = 0$
- $x^2 + 4x = -4$

Answers:

- $x = -9$ or $x = 1$
- $x = 0$ or $x = -4$
- $x = -2$

^{2}+ bx + c = 0