SOLUTION: Page 132 (problem 95): A length of wire 16 inches is to be cut into two pieces, and then each piece will be bent to form a square. Find the length of the two pieces if the sum of t

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Question 183085This question is from textbook Precalculus
: Page 132 (problem 95): A length of wire 16 inches is to be cut into two pieces, and then each piece will be bent to form a square. Find the length of the two pieces if the sum of the areas of the two squares is 10 square inches.
(x/4)^2 + ((16-x)/4)^2 = 10
I've gotten this far but can't figure out the next steps to solve it.
Thank you, Lynn Scott This question is from textbook Precalculus

Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
expanding __ (x^2)/16 + (256-32x+x^2)/16 = 10

multiplying by 16 and rearranging __ 2x^2-32x+256 = 160

subtracting 160 and dividing by 2 __ x^2-16x+48 = 0

factoring __ (x-12)(x-4) = 0

x equals 12 or 4 (which sum to 16)

one square is 9in^2 and the other is 1in^2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You're on the right track, you just need to solve the equation:


%28x%2F4%29%5E2+%2B+%28%2816-x%29%2F4%29%5E2+=+10+ Start with the given equation.


x%5E2%2F16+%2B+%28%2816-x%29%2F4%29%5E2+=+10+ Square x%2F4 to get %28x%2F4%29%5E2=%28x%2F4%29%28x%2F4%29=x%5E2%2F16


x%5E2%2F16+%2B+%2816-x%29%5E2%2F16+=+10+ Square %2816-x%29%2F4 to get %28%2816-x%29%2F4%29%5E2=%28%2816-x%29%2F4%29%28%2816-x%29%2F4%29=%2816-x%29%5E2%2F16


%28x%5E2%2B+%2816-x%29%5E2%29%2F16+=+10+ Combine the fractions.


x%5E2%2B+%2816-x%29%5E2+=+10%2816%29+ Multiply both sides by 16.


x%5E2%2B+%2816-x%29%5E2+=+160+ Multiply


x%5E2%2B+256-32x%2Bx%5E2+=+160+ FOIL


x%5E2%2B+256-32x%2Bx%5E2+-160+=0 Subtract 160 from both sides.


2x%5E2-32x%2B96=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-32, and c=96


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-32%29+%2B-+sqrt%28+%28-32%29%5E2-4%282%29%2896%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-32, and c=96


x+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%282%29%2896%29+%29%29%2F%282%282%29%29 Negate -32 to get 32.


x+=+%2832+%2B-+sqrt%28+1024-4%282%29%2896%29+%29%29%2F%282%282%29%29 Square -32 to get 1024.


x+=+%2832+%2B-+sqrt%28+1024-768+%29%29%2F%282%282%29%29 Multiply 4%282%29%2896%29 to get 768


x+=+%2832+%2B-+sqrt%28+256+%29%29%2F%282%282%29%29 Subtract 768 from 1024 to get 256


x+=+%2832+%2B-+sqrt%28+256+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2832+%2B-+16%29%2F%284%29 Take the square root of 256 to get 16.


x+=+%2832+%2B+16%29%2F%284%29 or x+=+%2832+-+16%29%2F%284%29 Break up the expression.


x+=+%2848%29%2F%284%29 or x+=++%2816%29%2F%284%29 Combine like terms.


x+=+12 or x+=+4 Simplify.


So the answers are x+=+12 or x+=+4


This means that the second length of the square is either

16-12=4 or 16-4=12

Note: either way, the two side lengths are 12 and 4


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Answer:

So the length of the two pieces are 12 and 4 inches.