Math

QuestionFind the width and length of a window by solving A(x)=x(x+4)=221A(x) = x(x+4) = 221. Simplify your answers.

Studdy Solution

STEP 1

Assumptions1. The quadratic function is A(x)=x(x+4)A(x) = x(x+4). The area of the window is given by A(x)=221A(x)=221
3. The length and width of the window are represented by xx and x+4x+4 respectively

STEP 2

First, we need to solve the equation A(x)=221A(x)=221 for xx. This means we set x(x+4)x(x+4) equal to 221221 and solve for xx.
x(x+4)=221x(x+4) =221

STEP 3

This equation is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c =0. To solve for xx, we can rewrite the equation in this form.
x2+x221=0x^2 +x -221 =0

STEP 4

To solve this quadratic equation, we can use the quadratic formula x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}.

STEP 5

Plug in the values for aa, bb, and cc into the quadratic formula.
x=4±(4)24(1)(221)2(1)x = \frac{-4 \pm \sqrt{(4)^2 -4(1)(-221)}}{2(1)}

STEP 6

implify the equation under the square root.
x=4±16+8842x = \frac{-4 \pm \sqrt{16 +884}}{2}

STEP 7

Further simplify the equation.
x=4±9002x = \frac{-4 \pm \sqrt{900}}{2}

STEP 8

Calculate the square root.
x=4±302x = \frac{-4 \pm30}{2}

STEP 9

We have two possible solutions for xx. Calculate both.
x=4+302=13x = \frac{-4 +30}{2} =13x2=4302=17x2 = \frac{-4 -30}{2} = -17

STEP 10

Since dimensions cannot be negative, we discard x2=17x2 = -17 and take x=13x =13 as the solution.

STEP 11

Now that we have the value for xx, we can find the values for the width and length of the window. The width is xx and the length is x+4x+4.
Width=x=13Width = x =13Length=x+4=13+4=17Length = x +4 =13 +4 =17The width of the window is13 units and the length is17 units.

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