Question 1194518
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x = width of the frame in centimeters.
This is some positive real number, meaning that x > 0


The width of 30 cm bumps up to 30+x+x = 30+2x cm
This is because we add on two copies of x for either side along the 30 cm dimension.
The same thing happens with the 20 cm bumping up to 20+2x cm.


The painting + frame has dimensions of (30+2x) cm by (20+2x) cm


Multiply those dimensions and set the product equal to the stated total area of 1200 sq cm.


length*width = area
(30+2x)*(20+2x) = 1200
600+60x+40x+4x^2 = 1200
600+100x+4x^2 = 1200
4x^2+100x+600-1200 = 0
4x^2+100x-600 = 0
4(x^2+25x-150) = 0
x^2+25x-150 = 0/4
x^2+25x-150 = 0


Now use the quadratic formula to find x.
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(25)+-sqrt((25)^2-4(1)(-150)))/(2(1))}}}


{{{x = (-25+-sqrt(1225))/(2)}}}


{{{x = (-25+-  35)/(2)}}}


{{{x = (-25+35)/(2)}}} or {{{x = (-25-35)/(2)}}}


{{{x = (10)/(2)}}} or  {{{x = (-60)/(2)}}}


{{{x = 5}}} or  {{{x = -30}}}
Ignore the negative solution. Recall at the top we stated that x > 0.


Answer: The width of the frame is 5 cm.
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