Creating Quadratic Equations for area and finding x

Question

A picture frame holds a 4-in. by 6-in. photograph. The frame adds a border x inches wide around three sides of the photo. On the fourth aide the frame forms a border that is 3x-0.5in wide. The combined area of the photograph and frame is 80.5 in^2. Write a quadratic equation for the combined area. Then use the quadratic formula to find x. 

Answer 1

Not happy with the question as it is not certain which side is the "fourth".  Is it a 4" side or a 6" side?

 

Assume it is a 4" side that means the 6" dimension will receive the extra 

x + 6 + (3x-0.5) =4x + 5.5

the other dimension will be

x + 4 + x = 2x + 4

80.5 = (4x+5.5)(2x+4) = 8x² + 27x + 22

0 = 8x² + 27x - 58.5

x = {-27 + √[27² - 4(8)(-58.5)]}/2(8)

note: the minus value will yield a negative length and is not a valid solution

x = [-27 + √(729 + 1872)]/16

x = (-27 + √2601)/16

x = (-27 + 51)/16  = 24/16 = 1.5 inches

 

BUT, if it is a 6" side then the 4" side gets the extra,

x + 4 + (3x-0.5) = 4x + 3.5

x + 6 + x = 2x + 6

80.5 = (4x+3.5)(2x+6) = 8x² + 31x +21

0 = 8x² + 31x - 59.5

x = {-31 + √[31² - 4(8)(-59.5)]}/2(8)

x = [-31 + √(961 + 1904)]/16

x= (-31 + √2865)/16

x = (-31 + 53.53)/16

x = 22.53/16 = 1.41 inches