Question 114393
First let L represent the length of the movie screen and let W represent the width of the screen.
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You are told that the length is 19 feet longer than the width. So you can say in equation form
that:
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L = W + 19
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You are also told that the area of the screen is 10416 sq ft. But the area of a rectangle
such as this screen is is found by multiplying its length L by its width W. So we can write
in equation form that:
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Area = L * W
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Substitute 10416 for A and W + 19 for L and you have:
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10416 = (W + 19)*W
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Multiply out the right side by multiplying W times each of the terms in the parentheses
and the equation becomes:
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10416 = W^2 + 19W
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Get this into standard quadratic form by subtracting 10416 from both sides to make the
equation:
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0 = W^2 + 19W - 10416
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Just to get it into a more conventional form, transpose the equation (switch sides) to
make it:
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W^2 + 19W - 10416 = 0
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This is in the standard quadratic form of:
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aW^2 + bW + c = 0
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and by comparing this standard form with the conventional for for this problem, we can see
that a = 1, b = 19, and c = -10416
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For the standard quadratic form, according to the quadratic formula the answers for the 
unknown are:
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{{{W = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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Substitute 1 for a, 19 for b, and -10416 for c and you have:
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{{{W = (-19 +- sqrt( 19^2-4*1*-10416 ))/(2*1) }}}
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Solve for the term under the radical. {{{19^2 = 361}}} and {{{-4*1*-10416 = 41664}}}. Add
these two terms to get that the term under the radical is {{{361 + 41664 = 42025}}}.
This reduces the answer to:
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{{{W = (-19 +- sqrt( 42025 ))/(2*1) }}}
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Taking the square root of 42025 on a calculator we get 205. Substituting this result further
reduces the equation for W to:
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{{{W = (-19 +- 205)/(2*1) }}}
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Multiply out the denominator ... 2*1 = 2 and this simplifies the equation to:
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{{{W = (-19 +- 205)/2 }}}
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The numerator must be positive so that the value of W is positive. [It would make no sense
to have a negative width.] Therefore, we can disregard the negative sign preceding the
205 and the equation becomes just:
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{{{W = (-19 + 205)/2 }}}
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The two terms in the numerator combine to 186 and the equation is simplified to:
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{{{W = 186/2 = 93}}}
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This tells you that the small dimension of the screen is 93 feet. The larger dimension
(the length) is 19 feet more or 93 + 19 = 112 ft.
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So the screen is 93 by 112 feet.
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Check ... the area of the screen is 93 * 112 = 10416 square feet, just as the problem
says it should be. Our answer checks.
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Hope this helps you to understand the problem and how to use the quadratic formula in
solving it.
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