Completing the Square

**Completing the square** is a technique to rewrite a quadratic expression $ax^2 + bx + c$ into the form $a(x - h)^2 + k$, called **vertex form**. The key insight: we add and subtract the same value to create a **perfect square trinomial**. $$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$ For example: $$x^2 + 6x + 9 = (x + 3)^2$$ Here, $\left(\frac{6}{2}\right)^2 = 9$ completes the square.

Basic Completing the Square

Complete the square: $x^2 + 8x$

Identify the coefficient of x: $b = 8$ = Coefficient is 8

Take half of b: $\frac{8}{2} = 4$ = Half is 4

Square the result: $4^2 = 16$ = Need to add 16

Add and subtract this value: $x^2 + 8x + 16 - 16$ = Balance maintained

Factor the perfect square trinomial: $(x + 4)^2 - 16$ = Completed square form

Answer: $x^2 + 8x = (x + 4)^2 - 16$

Solving by Completing the Square

Solve $x^2 + 6x + 2 = 0$ by completing the square

Move the constant to the right: $x^2 + 6x = -2$ = Isolated variable terms

Find the value to complete the square: $\left(\frac{6}{2}\right)^2 = 9$ = Need to add 9

Add to both sides: $x^2 + 6x + 9 = -2 + 9$ = $(x + 3)^2 = 7$

Take the square root of both sides: $x + 3 = \pm\sqrt{7}$ = Two solutions

Solve for x: $x = -3 \pm \sqrt{7}$ = Final answer

Answer: $x = -3 + \sqrt{7}$ or $x = -3 - \sqrt{7}$

Converting to Vertex Form

Write $y = x^2 - 4x + 7$ in vertex form and identify the vertex

Group the x terms: $y = (x^2 - 4x) + 7$ = Grouped

Complete the square inside parentheses: $\left(\frac{-4}{2}\right)^2 = 4$ = Add and subtract 4

Add and subtract inside, then regroup: $y = (x^2 - 4x + 4) + 7 - 4$ = Balance maintained

Factor and simplify: $y = (x - 2)^2 + 3$ = Vertex form

Identify the vertex: Vertex form: $y = (x - h)^2 + k$ = Vertex is $(2, 3)$

Answer: $y = (x - 2)^2 + 3$, vertex at $(2, 3)$

When the Leading Coefficient is Not 1

Complete the square: $2x^2 + 12x + 5$

Factor out the leading coefficient from x terms: $2(x^2 + 6x) + 5$ = Factor out 2

Complete the square inside: $\left(\frac{6}{2}\right)^2 = 9$ = Need 9 inside

Add and subtract 9 inside the parentheses: $2(x^2 + 6x + 9 - 9) + 5$ = Added 9, subtracted 9

Distribute the -9 and simplify: $2(x^2 + 6x + 9) - 18 + 5$ = Note: $2 \times (-9) = -18$

Factor and combine constants: $2(x + 3)^2 - 13$ = Final vertex form

Answer: $2x^2 + 12x + 5 = 2(x + 3)^2 - 13$

Completing the square is one of the most powerful algebraic techniques because it: - **Solves any quadratic equation** (even when factoring doesn't work) - **Reveals the vertex** of a parabola directly from the equation - **Derives the quadratic formula** (which comes from completing the square!) - **Transforms circles** from general to standard form in geometry Without this technique, many quadratic equations would be unsolvable by hand!

Mistake: Forgetting to add to both sides when solving an equation

Why: When completing the square in an equation, whatever you add to one side must be added to the other side to maintain equality.

Correct: After adding $\left(\frac{b}{2}\right)^2$ to the left, add the same value to the right.

Mistake: Not factoring out the leading coefficient first

Why: When $a \neq 1$ in $ax^2 + bx + c$, completing the square directly gives wrong results.

Correct: First factor out $a$ from the $x^2$ and $x$ terms, then complete the square inside.

Mistake: Using $b$ instead of $\frac{b}{2}$ to complete the square

Why: The formula requires half of the coefficient of $x$, squared.

Correct: Always use $\left(\frac{b}{2}\right)^2$, not $b^2$.

Mistake: Forgetting to multiply when distributing

Why: When $a$ is factored out, the value subtracted inside gets multiplied by $a$.

Correct: In $2(x^2 + 6x + 9 - 9)$, the $-9$ becomes $-18$ when distributed: $2(-9) = -18$.

Projectile Motion

Engineers use completing the square to find the maximum height of projectiles.

The height of a ball is $h = -16t^2 + 64t + 5$ feet. Completing the square: $h = -16(t - 2)^2 + 69$. Maximum height is 69 feet at $t = 2$ seconds.

Architecture and Design

Architects use completing the square to design parabolic arches and determine their highest points.

A bridge arch follows $y = -0.01x^2 + 2x$. Converting to vertex form: $y = -0.01(x - 100)^2 + 100$. The arch reaches 100 meters high at its center.

Economics and Profit Optimization

Businesses use quadratic models to find optimal pricing for maximum profit.

A company's profit is $P = -2x^2 + 40x - 150$ dollars where $x$ is the price. Vertex form: $P = -2(x - 10)^2 + 50$. Maximum profit of 50 dollars occurs at price 10 dollars.

Completing the square transforms $x^2 + bx$ into $(x + \frac{b}{2})^2 - (\frac{b}{2})^2$

The key value to add is always $\left(\frac{b}{2}\right)^2$

When $a \neq 1$, factor out $a$ from the $x$-terms first

Vertex form $a(x - h)^2 + k$ reveals the vertex $(h, k)$ directly

This technique can solve any quadratic equation, even when factoring fails

Q: When should I use completing the square instead of factoring?

A: Use completing the square when: (1) the quadratic doesn't factor nicely with integers, (2) you need to find the vertex of a parabola, or (3) you're deriving the quadratic formula. Factoring is faster when it works, but completing the square always works.

Q: Why do we add and subtract the same value?

A: Adding and subtracting the same value is like adding zero, which doesn't change the expression's value. This lets us create a perfect square trinomial without changing what the expression equals.

Q: How is this related to the quadratic formula?

A: The quadratic formula is derived by completing the square on the general form $ax^2 + bx + c = 0$. The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ comes directly from this process!

Exit Presentation

Completing the Square

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Algebra / Quadratic Equations

Completing the Square

Master the technique of completing the square to solve quadratic equations and rewrite them in vertex form.

Advanced30 min

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