Free Completing the Square Practice

Overview

Completing the square is a fundamental technique in high school algebra used to solve quadratic equations, rewrite quadratic functions, and prepare for more advanced topics. This page offers Completing the Square Practice problems with step‑by‑step examples, giving you a hands‑on way to learn the method. An online completing the square calculator is available alongside to verify your work and show each intermediate manipulation.

The General Method

Any quadratic equation can be written in the form

$$ax^2 + bx + c = 0 \qquad (a \neq 0).$$

The essence of the technique is to transform the expression $ax^2+bx+c$ into a perfect square trinomial plus a constant. Follow these steps:

1. Make the leading coefficient 1: If $a \neq 1$, divide the entire equation by $a$.
2. Isolate the variable terms: Move the constant term to the right–hand side: $$x^2 + \frac{b}{a}x = -\frac{c}{a}.$$
3. Add the square of half the coefficient of $x$: Compute $\left(\dfrac{b}{2a}\right)^2$ and add it to both sides: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}.$$
4. Factor the left side as a square: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}.$$
5. Take the square root: If the right side is non‑negative, $$\left|x + \frac{b}{2a}\right| = \sqrt{\frac{b^2 - 4ac}{4a^2}}.$$ Solving for $x$ yields the two real roots (or one repeated root). If the right side is negative, the equation has no real solutions, only complex ones.

Worked Examples

Example 1: A perfect square in disguise

Solve $x^2 - x + 0.25 = 1$.

Notice that $x^2 - x$ already contains the perfect square pattern: half of $-1$ is $-0.5$, and $(-0.5)^2 = 0.25$. Hence the left side is already $(x-0.5)^2$:

$$(x-0.5)^2 = 1 \quad\Longrightarrow\quad x-0.5 = \pm 1 \quad\Longrightarrow\quad x = 1.5 \;\text{or}\; x = -0.5.$$

Example 2: Leading coefficient not 1

Solve $2x^2 + 4x + 8 = 0$.

Divide both sides by 2:

$$x^2 + 2x + 4 = 0 \quad\Longrightarrow\quad x^2 + 2x = -4.$$

Complete the square by adding $\left(\frac{2}{2}\right)^2 = 1$:

$$x^2 + 2x + 1 = -4 + 1 \quad\Longrightarrow\quad (x+1)^2 = -3.$$

Because the right side is negative, there are no real roots. The complex solutions are $x = -1 \pm i\sqrt{3}$.

Example 3: A quadratic with no real solution

Solve $x^2 - 8x + 20 = 0$.

$$x^2 - 8x = -20 \quad\Longrightarrow\quad x^2 - 8x + 16 = -20 + 16 \quad\Longrightarrow\quad (x-4)^2 = -4.$$

Thus the equation has no real solutions; the complex solutions are $x = 4 \pm 2i$.

Practice Exercises

A. Complete the Square (Find the Missing Constant)

For each expression, determine the constant that must be added to form a perfect square trinomial. Write the resulting trinomial in factored form.

Expression	Constant to add	Completed trinomial	Factored form
$x^2 + 2x$	$1$	$x^2 + 2x + 1$	$(x+1)^2$
$x^2 - 6x$	$9$	$x^2 - 6x + 9$	$(x-3)^2$
$x^2 + 3x$	$\dfrac{9}{4}$	$x^2 + 3x + \dfrac{9}{4}$	$\left(x + \dfrac{3}{2}\right)^2$
$x^2 + 6x + 6$	$3$	$x^2 + 6x + 9$	$(x+3)^2$
$x^2 - 2x + 3$	$-2$	$x^2 - 2x + 1$	$(x-1)^2$

B. Solve Quadratic Equations by Completing the Square

Use the method to find all solutions (real or complex).

1. $x^2 + 8x - 9 = 0$
   Answer: $x = -9$ or $x = 1$

2. $3x^2 + 12x = 0$
   Answer: $x = 0$ or $x = -4$

3. $x^2 + 4x = -4$
   Answer: $x = -2$ (a double root)

Why This Method Matters

Completing the square is not only a tool for solving equations; it is also the key to rewriting a quadratic function in vertex form $y = a(x - h)^2 + k$. From this form you can read the vertex $(h,k)$ directly, which simplifies graphing and helps solve optimization problems. Mastering the technique now pays off throughout algebra, precalculus, and beyond.

Tips for Success

• Always ensure the $x^2$ coefficient is 1 before adding the square term.
• Add exactly the same quantity to both sides to keep the equation balanced.
• When the constant term on the right becomes negative, the equation has complex solutions; express them using $i = \sqrt{-1}$.
• Use the online step‑by‑step completing the square solver to check your work and see the process in action.