Question 204190
1)
Starting with:
{{{(8+2x)(4+2x) = 165}}} Expand  and rearrange the left side.
{{{4x^2+24x+32 = 165}}} Subtract 165 from both sides.
{{{4x^2+24x-133 = 0}}} Solve this quadratic equation using the quadratic formula:{{{x = (-b+-sqrt(b^2-4ac))/2a}}} where: a = 4, b = 24, and c = -133.
{{{x = (-24+-sqrt(24^2-4(4)(-133)))/2(4)}}}
{{{x = (-24+-sqrt(576-(-2128)))/8}}}
{{{x = (-24+-sqrt(2704))/8}}}
{{{x = (-24+52)/8}}} or {{{x = (-24-52)/8}}}
{{{highlight(x = 3.5)}}} or {{{cross(x = -9.5)}}} Discard the negative solution as the width of the flower bed can only be a positive quantity.
The width of the flower bed is 3.5 feet.
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2) Can {{{x(x-5) = 0}}} be solved by dividing both sides by x?
No, because you would get olny one of the solutions to this quadratic equation when, in fact, there are really two solutions!
Here's how you would solve it:
{{{x(x-5) = 0}}} Apply the zero product rule: If {{{a*b = 0}}} then either {{{a = 0}}} or {{{b = 0}}} or both.
{{{highlight(x = 0)}}} or {{{x-5 = 0}}}
If {{{x-5 = 0}}} then {{{highlight_green(x = 5)}}}
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3) Write a quadratic equation whose roots are:
{{{x = 5}}} and {{{x = -3}}} If these are the roots, then the factors are:
{{{x-5 = 0}}} and {{{x+3 = 0}}} Multiply these two factors together to get the original quadratic equation.
{{{y = (x-5)(x+3)}}} Multiply using FOIL. 
{{{y = x^2+3x-5x-15}}} Simplify.
{{{highlight(y = x^2-2x-15)}}}