Completing the Square

a

Let's examine the rectangle.

The rectangle has a width of $x$ feet and a length of $120 - 2 x .$ Area of a rectangle can be found by multiplying its width bcy its length. With this, we can write the equation that gives the area of the pen. $A = w \cdot l \Leftrightarrow A (x) = x (120 - 2 x)$

b

We can solve the equation by grouping.

A (x) = x (120 - 2 x)

Substitute

A (x) = 1512

1512 = x (120 - 2 x)

Simplify

DistrDistribute

x

1512 = 120 x - 2 x^{2}

SubEqn

LHS - 1512 = RHS - 1512

0 = 120 x - 2 x^{2} - 1512

RearrangeEqnRearrange equation

120 x - 2 x^{2} - 1512 = 0

CommutativePropAddCommutative Property of Addition

- 2 x^{2} + 120 x - 1512 = 0

DivEqn

LHS / - 2 = RHS / - 2

x^{2} - 60 x + 756 = 0

Factor

WriteDiffWrite as a difference

x^{2} - 42 x - 18 x + 756 = 0

FactorOutFactor out

x

x (x - 42) - 18 x + 756 = 0

FactorOutFactor out

- 18

x (x - 42) - 18 (x - 45) = 0

FactorOutFactor out

(x - 42)

(x - 42) (x - 18) = 0

Solve using the Zero Product Property

ZeroProdPropUse the Zero Product Property

x - 42 = 0 x - 18 = 0

AddEqn

(I):

LHS + 42 = RHS + 42

x = 42 x - 18 = 0

AddEqn

(II):

LHS + 18 = RHS + 18

x = 42 x = 18

As a result, the value of

x

must be

42 .

With this, we can find the dimensions of the pen.

Width : Length : x \Rightarrow 42 ft 120 - 2 x \Rightarrow 120 - 2 (42) = 36 ft

Completing the Square 1.9 - Solution