Question 568677
Let the width be {{{x}}} cm.
The length (in cm) is {{{x+9}}}.
The area of the rectangle is length time width. In square centimeters, it's
{{{x(x+9)=x^2+9x}}}
If that is 90 square centimeters,
{{{x^2+9x=90}}}
The expected quadratic equation answer is probably {{{highlight(x^2+9x-90=0)}}}, although {{{x^2+9x=90}}} and {{{x(x+9)=90}}} are equivalent equations.
To solve it, you could factor, complete the square, or use the quadratic formula.
Factoring, we see that {{{x^2+9x-90=(x+15)(x-6)}}} so the equation transforms into
{{{(x+15)(x-6)=0}}} with solutions {{{x=6}}} and {{{x=-15}}}.
Since the width of a frame is measured as a positive number, the only solution to the problem is
{{{highlight(x=6)}}}.
The width of the frame is 6 cm. The length is 15 cm ({{{6+9=15}}}).