Question 734058
{{{x}}}= length of the side of the square (in feet)
{{{x*x=196}}} --> {{{x^2=196}}} <--> {{{highlight(x^2-196=0)}}}
That is a quadratic equation.
 
If they have a real solution, quadratic equations can always be solved by using the quadratic formula.
 
If they have a real solution, quadratic equations can also always be solved by expressing them as {{{expression^2=number}}}. To get to that expression sometimes you have to "complete the square."
 
Some quadratic equations can be solved by factoring.
 
{{{x^2=196}}} <--> {{{x^2-196=0}}} can be solved using all the strategies a
listed above.
For example, knowing that {{{14^2=196}}} we figure that also {{{(-14)=196}}}
so the solutions to {{{x^2=196}}} are
{{{highlight(x=14)}}} and {{{highlight(x=-14)}}}
We did not even have to "complete the square."
Factoring is also easy if we know that {{{14^2=196}}} because we jnow that a difference of squares is a special product of the form
{{{a^2-b^2=(a+b)(a-b)}}} so
{{{x^2-196=x^2-14^2=(x+14)(x-14)}}} and we can re-write {{{x^2-196=0}}} as
{{{(x+14)(x-14)=0}}}
and if thatr product is zero, one of the factors must be zero, so
either {{{x+14=0}}} <--> {{{highlight(x=-14)}}}
or  {{{x-14=0}}} <--> {{{highlight(x=14)}}}
The quadratic formula applied to an equation of the form {{{ax^2+bx+c=0}}} says that the solutions (if they exist) are given by
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In the case of {{{x^2-196=0}}} , {{{a=1}}}, {{{b=0}}} and {{{c=-196}}} so
{{{x = (-0 +- sqrt(0^2-4*1*(-196)))/(2*1) }}} --> {{{x = (0 +- sqrt(4*196))/2 }}} --> {{{x = (0 +- 28)/2 }}}
which means {{{highlight(x=14)}}} or {{{highlight(x=-14)}}}