Understanding Quadratic Equations: Completing the Square and Perfect Square Trinomials

	$x + 3 = \pm 7$
Write as two equations	$x + 3 = 7$	$x + 3 = - 7$
Solve for $x$	$x = - 3 + 7$	$x = - 3 - 7$

x + 3 = \pm 7

Write as two equations

x + 3 = 7

x + 3 = - 7

Solve for

x

x = - 3 + 7

x = - 3 - 7

a The formula for the area of a rectangle will be used to create the quadratic equation representing the area of the pool.

A = ℓ w

In this formula,

A

represents the area,

ℓ

the length, and

w

the width of the rectangle. It is given that the area of the pool must be

768

square feet.

A = ℓ w Substitute 768 = ℓ w

Now, let

x

represent the width of the pool. Because the length will be

32

feet longer than the width, it can be represented as

x + 32 .

External credits: @kdekiara

By substituting the expressions for the length and width of the pool into the formula, the quadratic equation representing the area of the pool can be determined.

768 = ℓ w

SubstituteII

$ℓ = x + 32$ , $w = x$

768 = (x + 32) \cdot x

Simplify

Distr

Distribute $x$

768 = x^{2} + 32 x

RearrangeEqn

Rearrange equation

x^{2} + 32 x = 768

b To solve the quadratic equation written in Part A, the method of completing square will be used. Notice that the equation is already in the form

x^{2} + b x = c .

x^{2} + 32 x = 768

To complete the square on the left-hand side of the equation,

(\frac{b}{2})^{2}

needs to be calculated. In this case,

b = 32 .

(\frac{3 2}{2})^{2} \Leftrightarrow 1 6^{2}

This term needs to be added to both sides of the equation to produce an equivalent equation.

x^{2} + 32 x + 1 6^{2} = 768 + 1 6^{2}

The perfect square trinomial can now be factored.

x^{2} + 32 x + 1 6^{2} = 768 + 1 6^{2}

SplitIntoFactors

Split into factors

x^{2} + 2 \cdot x \cdot 16 + 1 6^{2} = 768 + 1 6^{2}

FacPosPerfectSquare

$a^{2} + 2 a b + b^{2} = (a + b)^{2}$

(x + 16)^{2} = 768 + 1 6^{2}

CalcPow

Calculate power

(x + 16)^{2} = 768 + 256

AddTerms

Add terms

(x + 16)^{2} = 1024

Taking the square root of both sides of the equation will produce two linear equations whose solutions are also solutions of the quadratic equation.

(x + 16)^{2} = 1024

SqrtEqn

$LHS = RHS$

(x + 16)^{2} = 1024

$a^{2} = \pm a$

x + 16 = \pm 1024

CalcRoot

Calculate root

x + 16 = \pm 32

This can be separated into two equations.

$x + 16 = \pm 32$
$x + 16 = 32$	$x + 16 = - 32$
$x = 16$	$x = - 48$

The quadratic equation has two solutions, one negative and one positive. Because the dimensions of the pool cannot be negative, the negative solution is not an option for Vincenzo. Therefore, the dimensions of the pool are given by the positive solution.

Width : Length : 16 ft^{2} 16 + 32 = 48 ft^{2}

$x + 1 = \pm 5$
$x + 1 = 5$	$x + 1 = - 5$
$x = 4$	$x = - 6$

x + 1 = \pm 5

x + 1 = 5

x + 1 = - 5

x = 4

x = - 6

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Catch-Up and Review

Challenge

Is It Possible to Rewrite Equations That Are in Standard Form to Find Their Solutions?

Discussion

Completing the Square

Pop Quiz

Practicing Finding the Values for Completing the Square

Discussion

Extending Completing the Square to Quadratic Equations

Method

Solving a Quadratic Equation by Completing the Square

Example

Solving Quadratic Equations by Completing the Square

Answer

Hint

Solution

Example

Completing the Square to Solve Daily Problems

Hint

Solution

Example

Completing the Square to Identify Quadratic Equations With No Solutions

Answer

Hint

Solution

Closure

Completing the Square to Decipher the Code

Hint

Solution

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