SAT — mathHard5 min read

Completing the Square

Master completing the square for the Digital SAT. Convert quadratics to vertex form and solve equations that don't factor neatly.

Tutor AI Team2026-02-25T01:23:11.204Z

Completing the square transforms a quadratic expression from standard form $ax^2 + bx + c$ into vertex form $a(x - h)^2 + k$ . This technique is valuable for finding the vertex of a parabola, solving quadratic equations that don't factor easily, and deriving the quadratic formula itself. The Digital SAT tests this skill in both directions.

Core Concepts

The Process (when $a = 1$ )

To complete the square for $x^2 + bx$ :

Take half of $b$ : $\frac{b}{2}$
Square it: $\left(\frac{b}{2}\right)^2$
Add and subtract this value.

$x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$

Example: $x^2 + 6x + 5$

Half of 6 = 3. Square: 9.

$= (x^2 + 6x + 9) - 9 + 5 = (x + 3)^2 - 4$

When $a \neq 1$

Factor out $a$ from the first two terms first:

$2x^2 + 12x + 7 = 2(x^2 + 6x) + 7 = 2(x^2 + 6x + 9 - 9) + 7 = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$

Solving Equations by Completing the Square

Example: Solve $x^2 + 8x + 10 = 0$

$x^2 + 8x = -10$

Add $(8/2)^2 = 16$ to both sides: $x^2 + 8x + 16 = 6$

$(x + 4)^2 = 6$

$x + 4 = \pm\sqrt{6}$

$x = -4 \pm \sqrt{6}$

Vertex Form

$a(x - h)^2 + k$ tells you the vertex is at $(h, k)$ .

From the example: $(x + 3)^2 - 4$ → vertex at $(-3, -4)$ .

Be careful with signs: $(x + 3)^2 = (x - (-3))^2$ , so $h = -3$ .

Strategy Tips

Tip 1: "Half and Square" Is the Key Step

Take half the coefficient of $x$ , square it. This number completes the square.

Tip 2: Add and Subtract the Same Value

When completing the square within an expression (not solving an equation), add and subtract to keep the expression equivalent.

Tip 3: When $a \neq 1$ , Factor $a$ Out First

This is the most common source of errors. Factor $a$ from the $x^2$ and $x$ terms before completing the square.

Tip 4: Use This to Find the Vertex

The SAT often asks for the minimum/maximum value of a quadratic. Completing the square gives the vertex directly.

Tip 5: Connect to the Quadratic Formula

Completing the square on $ax^2 + bx + c = 0$ produces the quadratic formula. Understanding this connection deepens your mastery.

Worked Example: Example 1

Problem

Write $x^2 - 10x + 21$ in vertex form.

Half of $-10$ : $-5$ . Square: 25.

$= (x^2 - 10x + 25) - 25 + 21 = (x - 5)^2 - 4$

Vertex: $(5, -4)$

Solution

Worked Example: Example 2

Problem

Solve $x^2 + 4x - 1 = 0$ .

$x^2 + 4x = 1$

$(x + 2)^2 = 5$

$x = -2 \pm \sqrt{5}$

Solution

Worked Example: Example 3

Problem

What is the minimum value of $f(x) = x^2 - 6x + 13$ ?

$= (x - 3)^2 + 4$

Minimum value is $4$ (at $x = 3$ ).

Solution

Worked Example: SAT-Style

Problem

The function $f(x) = 2x^2 - 8x + 3$ can be written as $f(x) = 2(x - h)^2 + k$ . What is $k$ ?

$= 2(x^2 - 4x) + 3 = 2(x^2 - 4x + 4 - 4) + 3 = 2(x - 2)^2 - 8 + 3 = 2(x - 2)^2 - 5$

$k = -5$

Solution

Worked Example: Example 5

Problem

Solve $2x^2 + 12x + 14 = 0$ .

Divide by 2: $x^2 + 6x + 7 = 0$

$(x + 3)^2 = 2$

$x = -3 \pm \sqrt{2}$

Solution

Practice Problems

Problem 1

Write $x^2 + 8x + 12$ in vertex form.

Problem 2

Solve $x^2 - 2x - 4 = 0$ by completing the square.

Problem 3

Find the minimum value of $f(x) = x^2 + 12x + 40$ .

Problem 4

Write $3x^2 - 18x + 25$ in the form $a(x-h)^2 + k$ .

Problem 5

The vertex of $f(x) = x^2 + bx + 10$ is at $x = -3$ . Find $b$ .

Problem 6

Solve $x^2 - 6x + 2 = 0$ . Leave your answer in exact form.

Want to check your answers and get step-by-step solutions?

Common Mistakes

Forgetting to subtract what you added. If you add 9 inside the expression, subtract 9 outside (or account for it).
Sign errors with the vertex. $(x + 3)^2$ means $h = -3$ , not $h = 3$ .
Not factoring out $a$ first. When $a \neq 1$ , you must factor out $a$ before completing the square.
Distributing $a$ incorrectly. After completing the square inside the parentheses, remember to multiply the subtracted constant by $a$ .
Taking only the positive root. $x^2 = k$ gives $x = \pm\sqrt{k}$ .

Frequently Asked Questions

When should I use completing the square vs. the quadratic formula?

Use completing the square when you need the vertex form, or when the question explicitly asks for it. Use the formula for direct solving.

Can completing the square give complex solutions?

If $(x - h)^2 = k$ and $k < 0$ , there are no real solutions. The SAT typically stays with real numbers.

Is this tested directly on the SAT?

Yes — both "write in vertex form" and "find the minimum/maximum" questions require this technique.

What's the vertex formula shortcut?

The x-coordinate of the vertex is $x = -\frac{b}{2a}$ . Then $y = f\left(-\frac{b}{2a}\right)$ . This is equivalent to completing the square.

Do I need to memorise the process?

Yes. "Take half the coefficient of $x$ , square it" is the core step.

Key Takeaways

✓
Process: take half the $x$ -coefficient, square it, add and subtract.
✓
Vertex form: $a(x-h)^2 + k$ has vertex $(h, k)$ .
✓
For solving: move the constant, complete the square, take $\pm$ square root.
✓
When $a \neq 1$ : factor out $a$ first, then complete the square.
✓
Minimum/maximum value of a quadratic is the $k$ in vertex form.
✓
The vertex shortcut $x = -\frac{b}{2a}$ is a quick alternative.

Tags:

#sat#math#advanced-math#quadratics#completing-the-square

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Core Concepts

The Process (when a=1a = 1a=1)

When a≠1a \neq 1a=1

Solving Equations by Completing the Square

Vertex Form

Strategy Tips

Tip 1: "Half and Square" Is the Key Step

Tip 2: Add and Subtract the Same Value

Tip 3: When a≠1a \neq 1a=1, Factor aaa Out First

Tip 4: Use This to Find the Vertex

Tip 5: Connect to the Quadratic Formula

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?

The Process (when $a = 1$ )

When $a \neq 1$

Tip 3: When $a \neq 1$ , Factor $a$ Out First