Question 1205624: Lawrence is increasing his rectangular patio in his backyard. His patio is currently 12 feet by 10 feet. He wants to increase the patio by adding a decorative tile the same width (x) all the way around creating a total area of 180 square feet. Select all the quadratic equations that represent Lawrence’s new patio area.
A x^2 + 11x − 15 = 0
B x^2 − 11x − 15 = 0
C x^2 + 22x − 60 = 0
D x^2 − 22x − 60 = 0
E 4x^2 + 44x − 60 = 0
F 4x^2 − 44x − 60 = 0
Found 4 solutions by josgarithmetic, MathLover1, math_tutor2020, ikleyn:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! If this unknown border width is assigned as x, then Total Area is accounted as
;
work with that and finish your answer.
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Choice A.
Answer by MathLover1(20849)  (Show Source):
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
x = width of the decorative tile
x is some real number such that x > 0
Diagram
We have a 12 by 10 rectangle inside a 12+2x by 10+2x rectangle.
The 2x refers to adding 2 copies of x (one on either side) for each length and width.
Then,
area = (length)*(width)
area = (12+2x)*(10+2x)
area = w*(10+2x) .......... let w = 12+2x
area = 10w+2xw
area = 10( w )+2x( w )
area = 10( 12+2x )+2x( 12+2x ) ..... plug in w = 12+2x
area = 120+20x+24x+4x^2
area = 4x^2 + 44x + 120
The box method or the FOIL rule are alternative methods to get the expression shown above.
Set this equal to the desired area of 180 square feet and we get:
area = 180
4x^2 + 44x + 120 = 180
4x^2 + 44x + 120 - 180 = 0
4x^2 + 44x - 60 = 0
which is one possible answer.
We could divide both sides by 4 to get:
4x^2 + 44x - 60 = 0
(4x^2 + 44x - 60)/4 = 0/4
(4x^2)/4 + (44x)/4 - 60/4 = 0
x^2 + 11x - 15 = 0
which is another possible answer.
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Answers: choice A and choice E
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
It is incorrect to ask which of these equations represents the area.
No one of these equations, literally speaking, represents the area.
Mathematically correctly is to say that equations A and E are correct equations
for the width x of the added strip.
Yes, some of these equations originate from the equation for the area,
but after transformations, they just do not represent the area.
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Please don't think that I'm picking on words.
My writing is just the CORRECT mathematical verbal formulation.
Don't curse me, just learn from me.
It is why and for what you come to this forum. ( <<<--- just in case if you do not understand for what and why you come here).
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