8.7Completing the Square
Completing the square transforms any quadratic into a perfect square trinomial — revealing the vertex form and providing a universal method for solving quadratics that works even when factoring fails.
Why This Matters
Completing the square is how the quadratic formula was derived — understanding it gives you deep insight into quadratic structure. It's also used to convert to vertex form and to derive the equation of a circle.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does completing the square convert a quadratic equation into vertex form, and how does this method connect to the quadratic formula?
Lesson Overview
Completing the square is a technique that rewrites any quadratic expression as a perfect square trinomial. For a=1: move the constant to the right, add (b/2)² to both sides, factor the left side as (x + b/2)², then take the square root and solve. When a ≠ 1, divide every term by a first. This method always works, reveals the vertex form y = a(x−h)² + k, and — when applied to the general form ax²+bx+c=0 — produces the quadratic formula itself.
Completing the Square — Step-by-Step Process
Geometric Interpretation — Why It Works
x² — the large purple square has area x².
Two rectangles — each has area x·(b/2), representing the bx term split in half.
Missing corner — adding (b/2)² completes the big square, giving (x + b/2)².
x² + bx + (b/2)² = (x + b/2)²
Worked Examples
Solve x² + 6x − 7 = 0 by completing the square
Move constant: x² + 6x = 7
(b/2)² = (6/2)² = 9. Add to both sides: x² + 6x + 9 = 16
Factor: (x + 3)² = 16
Take square root: x + 3 = ±4
x = 1 or x = −7
Solve x² − 4x − 5 = 0 by completing the square
Move constant: x² − 4x = 5
(b/2)² = (−4/2)² = 4. Add to both sides: x² − 4x + 4 = 9
Factor: (x − 2)² = 9
Take square root: x − 2 = ±3
x = 5 or x = −1
Solve x² + 2x + 5 = 0 by completing the square
Move constant: x² + 2x = −5
(b/2)² = 1. Add to both sides: x² + 2x + 1 = −4
Factor: (x + 1)² = −4
Take square root: x + 1 = ±√(−4)
D < 0 → no real solutions
Convert y = x² − 8x + 3 to vertex form
Group x-terms: y = (x² − 8x) + 3
(b/2)² = 16. Add and subtract inside: y = (x² − 8x + 16) + 3 − 16
Factor: y = (x − 4)² − 13
Vertex: (4, −13)
Solve 2x² − 12x + 10 = 0 by completing the square
Divide by 2: x² − 6x + 5 = 0
Move constant: x² − 6x = −5
(b/2)² = 9. Add to both sides: x² − 6x + 9 = 4
Factor: (x − 3)² = 4
Take square root: x − 3 = ±2
x = 5 or x = 1
Guided Practice
Guided Practice Video: Completing the Square
Review completing the square to solve quadratic equations, including the a≠1 case, before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve x² + 4x − 12 = 0 by completing the square.
Hint: Move −12: x²+4x=12. Add (4/2)²=4 to both sides. Factor and solve.
Solve x² − 10x + 9 = 0 by completing the square.
Hint: Move 9: x²−10x=−9. Add (−10/2)²=25 to both sides. (x−5)²=16.
Convert y = x² + 4x − 1 to vertex form.
Hint: Complete the square: add and subtract (4/2)²=4. y=(x+2)²−5.
Solve 3x² + 6x − 9 = 0 by completing the square.
Hint: Divide by 3 first: x²+2x−3=0. Move −3: x²+2x=3. Add 1 to both sides.
Solve x² − 6x + 10 = 0 by completing the square. What do you notice?
Hint: Move 10: x²−6x=−10. Add 9: (x−3)²=−1. What does a negative right side mean?
Key Vocabulary
Completing the Square
A method of rewriting a quadratic by adding (b/2)² to create a perfect square trinomial.
Example: x²+6x+9 = (x+3)²
Perfect Square Trinomial
A trinomial of the form x²+bx+(b/2)² that factors as (x+b/2)².
Example: x²−8x+16 = (x−4)²
Vertex Form
y = a(x−h)²+k, where (h, k) is the vertex of the parabola. Obtained by completing the square.
(b/2)² Method
The key step in completing the square: take half the coefficient of x, then square it to find the number to add to both sides.
Practice Questions
Interactive Practice — 5 Questions
To complete the square for x² + 8x, what number must be added?
Which is the correct vertex form of y = x² − 6x + 2?
Solve x² + 6x − 7 = 0 by completing the square.
What is the first step when completing the square for 2x² − 8x + 3 = 0?
Completing the square on ax²+bx+c=0 produces which formula?
Independent Practice
Independent Practice
Solve by completing the square: x² + 8x + 7 = 0
Solve by completing the square: x² − 2x − 8 = 0
Convert to vertex form: y = x² + 6x + 11
Solve by completing the square: 2x² + 4x − 6 = 0
Challenge: Derive the quadratic formula by completing the square on ax² + bx + c = 0.
ChallengeCommon Mistakes
Adding (b/2)² to only one side of the equation.
Whatever you add to the left side, you must add to the right side too. Both sides must stay equal.
Forgetting to divide by a before completing the square when a ≠ 1.
When a ≠ 1, divide every term (including the constant) by a first, then proceed with the (b/2)² method.
Using b instead of b/2 — e.g., for x²+6x, adding 6² = 36 instead of 3² = 9.
The number to add is (b/2)², not b². Take HALF of b first, then square it.
Forgetting the ± when taking the square root of both sides.
√(k) = ±√k. Both the positive and negative roots must be considered to get both solutions.
Math Tips
The (b/2)² shortcut: for x²+bx, the number to add is always (half the x-coefficient) squared.
Completing the square is the most powerful method for converting to vertex form — use it whenever you need the vertex of a parabola.
If the right side becomes negative after completing the square, the equation has no real solutions (D < 0).
Connection: completing the square on ax²+bx+c=0 step-by-step produces the quadratic formula — understanding this derivation is a common AP and honors exam question.