Answer:
The quadratic equation that represents this problem is
4x² + 56x - 165 = 0
Explanation:
The dimensions of the picture are 16 length and 12 width both in inches
The area of the picture without frame = 16 x 12 = 192 square inches
Let the frame have a width of x inches. The frame surrounds the picture on both sides of the length and both sides of the breadth
Then the length with the frame = x + 16 + x = 2x + 16 inches
The width of the frame = x + 12 + x = 2x + 12 inches
The area of the painting with the frame
= (2x + 16)(2x + 12)
Using the FOILmethod to multiply these two expressions we get
(2x + 16)(2x + 12) = (2x)(2x) + 2x(12) + 16(2x) + 16 x 12
= 4x² + 24x + 32x + 192
= 4x² + 56x + 192
The area of the frame by itself is the area with frame - area of picture alone
Area of frame = 4x² + 56x + 192 - 192
=> 4x² + 56x
We are given the area of frame is 165 square inches
Hence we get
4x² + 56x = 165
To represent as a quadratic equation move 156 from the right side to the left side to get 0 on the right side
4x² + 56x - 165= 0
ANSWER