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2=\frac{\left(12+h\right)^{2}}{12^{2}}
Anything divided by one gives itself.
2=\frac{144+24h+h^{2}}{12^{2}}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+h\right)^{2}.
2=\frac{144+24h+h^{2}}{144}
Calculate 12 to the power of 2 and get 144.
2=1+\frac{1}{6}h+\frac{1}{144}h^{2}
Divide each term of 144+24h+h^{2} by 144 to get 1+\frac{1}{6}h+\frac{1}{144}h^{2}.
1+\frac{1}{6}h+\frac{1}{144}h^{2}=2
Swap sides so that all variable terms are on the left hand side.
1+\frac{1}{6}h+\frac{1}{144}h^{2}-2=0
Subtract 2 from both sides.
-1+\frac{1}{6}h+\frac{1}{144}h^{2}=0
Subtract 2 from 1 to get -1.
\frac{1}{144}h^{2}+\frac{1}{6}h-1=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
h=\frac{-\frac{1}{6}±\sqrt{\left(\frac{1}{6}\right)^{2}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{144} for a, \frac{1}{6} for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}
Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.
h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-\frac{1}{36}\left(-1\right)}}{2\times \frac{1}{144}}
Multiply -4 times \frac{1}{144}.
h=\frac{-\frac{1}{6}±\sqrt{\frac{1+1}{36}}}{2\times \frac{1}{144}}
Multiply -\frac{1}{36} times -1.
h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{18}}}{2\times \frac{1}{144}}
Add \frac{1}{36} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{2\times \frac{1}{144}}
Take the square root of \frac{1}{18}.
h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}}
Multiply 2 times \frac{1}{144}.
h=\frac{\sqrt{2}-1}{\frac{1}{72}\times 6}
Now solve the equation h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}} when ± is plus. Add -\frac{1}{6} to \frac{\sqrt{2}}{6}.
h=12\sqrt{2}-12
Divide \frac{-1+\sqrt{2}}{6} by \frac{1}{72} by multiplying \frac{-1+\sqrt{2}}{6} by the reciprocal of \frac{1}{72}.
h=\frac{-\sqrt{2}-1}{\frac{1}{72}\times 6}
Now solve the equation h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}} when ± is minus. Subtract \frac{\sqrt{2}}{6} from -\frac{1}{6}.
h=-12\sqrt{2}-12
Divide \frac{-1-\sqrt{2}}{6} by \frac{1}{72} by multiplying \frac{-1-\sqrt{2}}{6} by the reciprocal of \frac{1}{72}.
h=12\sqrt{2}-12 h=-12\sqrt{2}-12
The equation is now solved.
2=\frac{\left(12+h\right)^{2}}{12^{2}}
Anything divided by one gives itself.
2=\frac{144+24h+h^{2}}{12^{2}}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+h\right)^{2}.
2=\frac{144+24h+h^{2}}{144}
Calculate 12 to the power of 2 and get 144.
2=1+\frac{1}{6}h+\frac{1}{144}h^{2}
Divide each term of 144+24h+h^{2} by 144 to get 1+\frac{1}{6}h+\frac{1}{144}h^{2}.
1+\frac{1}{6}h+\frac{1}{144}h^{2}=2
Swap sides so that all variable terms are on the left hand side.
\frac{1}{6}h+\frac{1}{144}h^{2}=2-1
Subtract 1 from both sides.
\frac{1}{6}h+\frac{1}{144}h^{2}=1
Subtract 1 from 2 to get 1.
\frac{1}{144}h^{2}+\frac{1}{6}h=1
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{\frac{1}{144}h^{2}+\frac{1}{6}h}{\frac{1}{144}}=\frac{1}{\frac{1}{144}}
Multiply both sides by 144.
h^{2}+\frac{\frac{1}{6}}{\frac{1}{144}}h=\frac{1}{\frac{1}{144}}
Dividing by \frac{1}{144} undoes the multiplication by \frac{1}{144}.
h^{2}+24h=\frac{1}{\frac{1}{144}}
Divide \frac{1}{6} by \frac{1}{144} by multiplying \frac{1}{6} by the reciprocal of \frac{1}{144}.
h^{2}+24h=144
Divide 1 by \frac{1}{144} by multiplying 1 by the reciprocal of \frac{1}{144}.
h^{2}+24h+12^{2}=144+12^{2}
Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
h^{2}+24h+144=144+144
Square 12.
h^{2}+24h+144=288
Add 144 to 144.
\left(h+12\right)^{2}=288
Factor h^{2}+24h+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(h+12\right)^{2}}=\sqrt{288}
Take the square root of both sides of the equation.
h+12=12\sqrt{2} h+12=-12\sqrt{2}
Simplify.
h=12\sqrt{2}-12 h=-12\sqrt{2}-12
Subtract 12 from both sides of the equation.