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Rational Expressions

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a)     The area of a rectangle is x^2+7x+13.  If the length is (x+4), write an expression for the width of the rectangle, including the remainder.

b)      If the perimeter of the rectangle is 7, what are the length and width?

Whiz333  Jun 5, 2017

#1
+7324
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a)

area = length * width

x2 + 7x + 13  =  (x + 4) * width                 Divide both sides of the equation by (x + 4) .

(x2 + 7x + 13)/(x + 4)  =  width                 We can do this using long division....

So we get that

width  =  $$x+3+\frac1{x+4}$$

b)

perimeter = 2[ length  +  width ]

7 = 2[  x + 4  +  x + 3 + $$\frac1{x+4}$$  ]              Divide both sides by 2 .

3.5 = 2x + 7 + $$\frac1{x+4}$$                               Subtract 7 from both sides.

-3.5 = 2x + $$\frac1{x+4}$$                                    Multiply through by ( x + 4 )

-3.5(x + 4) = 2x(x + 4) + 1

-3.5x - 14 = 2x2 + 8x + 1                      Add  3.5x  and  14  to both sides.

0 = 2x2 + 11.5x + 15                             We can use the quadratic formula to solve for x.

$$x = {-11.5 \pm \sqrt{132.25-120} \over 4}={-11.5 \pm 3.5 \over 4}=-2.785\pm0.875$$

So, we get that  x = -2     and     x = -3.75

So......if x = -2

length = -2 + 4    =    2

width  = -2 + 3 + 1/(-2+4) = 1 + 1/2    =    1.5

And....if x = -3.75

length = -3.75 + 4    =    0.25

width  = -3.75 + 3 + 1/(-3.75+4) = -0.75 + 4    =    3.25

hectictar  Jun 5, 2017
#1
+7324
+1

a)

area = length * width

x2 + 7x + 13  =  (x + 4) * width                 Divide both sides of the equation by (x + 4) .

(x2 + 7x + 13)/(x + 4)  =  width                 We can do this using long division....

So we get that

width  =  $$x+3+\frac1{x+4}$$

b)

perimeter = 2[ length  +  width ]

7 = 2[  x + 4  +  x + 3 + $$\frac1{x+4}$$  ]              Divide both sides by 2 .

3.5 = 2x + 7 + $$\frac1{x+4}$$                               Subtract 7 from both sides.

-3.5 = 2x + $$\frac1{x+4}$$                                    Multiply through by ( x + 4 )

-3.5(x + 4) = 2x(x + 4) + 1

-3.5x - 14 = 2x2 + 8x + 1                      Add  3.5x  and  14  to both sides.

0 = 2x2 + 11.5x + 15                             We can use the quadratic formula to solve for x.

$$x = {-11.5 \pm \sqrt{132.25-120} \over 4}={-11.5 \pm 3.5 \over 4}=-2.785\pm0.875$$

So, we get that  x = -2     and     x = -3.75

So......if x = -2

length = -2 + 4    =    2

width  = -2 + 3 + 1/(-2+4) = 1 + 1/2    =    1.5

And....if x = -3.75

length = -3.75 + 4    =    0.25

width  = -3.75 + 3 + 1/(-3.75+4) = -0.75 + 4    =    3.25

hectictar  Jun 5, 2017