super hard algebra, prove your math skills!!?

Question

Solve the verbal problem with a quadratic equation

1. The length of a rectangle is 3 inches more than twice its width, and its area is 65 square inches. Find the dimensions. length = 2w+3, width = w.

2. The width of a rectangle is two-thirds of its length, and its area is 216 square meters. Find the dimensions. length = L, width = 2/3L

Answer 1

1) Because area = lw, this gives the equation 'w(2w + 3) = 65'. Distributing gives 2w^2 + 3w - 65 = 0

The Quadratic Formula is {-b ± sqrt(b^2 - 4ac)}/2a.

a = coefficient of w^2, 2

b = coefficient of w, 3

c = the constant, -65

Now you have everything you need to do this on your own.



2) L(2/3L) = 216. 2/3L^2 - 216 = 0. Now repeat the steps shown in problem 1, knowing that b will be 0 because there is no x.

Answer 2

1)

area=65=(2w+3)*w=2w2+3w implies 2w2+3w-65=0



(2w + 13)(w - 5)=0 implies 2w+13=0 implies w= - 13/2 impossible



or w-5 =0 implies w=5 inches and L = 13 inches



2)

216 = L*2/3L=2/3L2 implies L2 = 3*216/2=324 IMPLIES

L= 18 and w =12

Answer 3

1) w*(2w+3)=65 ------ 2w^2 +3w-65=0 you will need to do the Pythagorean theorem for this one. I don't really feel like doing it but it is just plugging in numbers.



2) 2/3L*L=216 ------- 2/3L^2=216 ----- L^2=324-------- L=18 meters^2

Answer 4

area = width*length

area = 65 in^2 = (2w + 3)*w

2w^2 + 3w = 65

w + (3/2)w = 65/2

(w + 3/4)^2 = 529/16

w = 5.75 - 0.75 = 5 in therefore

length = 2(5) + 3 = 13

area = 65 = width*length = 5*13



The second one is done the same way

Answer 5

1. 65=(2w+3)w

65=2w^2+3w

0=2w^2+3w-65

0=(w-5)(2w+13)

w=5

2w=-13, w= -13/2 (width can't be negative so this value is not the answer.

Therefore, w=5 in

l=2w+3

l=2(5)+3

l=13 in



2. w=2/3L, A=lw

216=L(2/3L)

216=2/3L^2

2/3L^2-216=0

3/2(2/3L^2-216)=3/2(0)

L^2-324=0

L^2=324

L=√324

L=18 m



w=2/3L

w=2/3(18)

w=12m

Answer 6

1. 2w^2 + 3w -65 = 0

(w-5)(2w+13) = 0

w = 5

l = 13



2. 2L^2/3 = 216

L^2 = 3*216/2 = 144

L = 18

W = 12

Answer 7

1. L = 2w+3



2(2w+3) +2 (w) =65

4w +6 +2w =65

6w = 59

w= 59/6 or 9-5/6