how to solve word problems by factoring

Question

The length of a rectangle is 3cm more than twice the width. The area of the rectangle is 90cm squared. Find the dimensions of the rectangle. 

Answer 1

First, translate the words to concise math expressions.

 

Assign L = length; W = width

 

"The length of a rectangle is 3cm more than twice the width"   means  L = 3+2*W     [eq1]

"The area of the rectangle is 90cm squared"     means   Area = L*W = 90        [eq2]

"Find the dimensions of the rectangle"    means     report  L and W

 

O.K., now the math:

    L*W = 90       [eq2]

    (3+2*W)*W = 90     [putting value for L from eq1 into eq2]

     3W + 2*W^2 = 90     [distribute W)

     2*W^2 +3W - 90 = 0    [rearrange terms to get equation that equals 0]

    (W-6)*(2W+15) = 0      (find factors)

  So, either W=6  or W=-7.5     (choose positive width)

 

Put W=6 into eq2 to get L*6 = 90,  so L=15

 

Check:

    Is 15  3 more than twice 6 ?    yes

    Is the area (15*6) equal to 90 ?    yes

Answer 2

Lets say that the width of the rectangle is x and length is y.

 

length is 3cm more than twice the width --> y = 2x + 3

 

Area is 90cm sq --> x*y = 90

 

Substitute y from first equation into second --> x*(2x+3) = 90

--> 2x² + 3x = 90

--> 2x² + 3x - 90 = 0

--> 2x² + 15x - 12x - 90 = 0            (since 3x = 15x -12x and 2*90 = 15*12) <- factoring rule

--> x(2x + 15) - 6(2x + 15) = 0       (factor x from first two terms and 6 from last two.

--> (x - 6)(2x + 15) = 0                   (distributive property)

 

--> x - 6 = 0 or 2x + 15 = 0

--> x = 6 or x = -15/2

 

Since length/width cannot be negative, x = 6

 

y = 2x + 3 = 2*6 + 3 = 15