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

Multiply the -9 by 2 and take it out of the parentheses: $2(x^2 + 6x + 9) - 18 + 5$ = 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 a thrown object.

The height of a ball is $h = -5t^2 + 10t + 1.5$ meters. Completing the square: $h = -5(t - 1)^2 + 6.5$. The maximum height is 6.5 meters at $t = 1$ second.

Architecture and Design

Architects use completing the square to design parabolic arches and find their highest point.

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

Economics and Profit Optimization

Businesses use quadratic models to find the price that gives the maximum profit.

A company's profit is $P = -2x^2 + 40x - 150$ dollars, where $x$ is the price in dollars. Vertex form: $P = -2(x - 10)^2 + 50$. The maximum profit of 50 dollars comes at a price of 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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