Solve 2/1=(12+h)^2/12^2

$$\frac { 2 } { 1 } = \frac { ( 12 + h ) ^ { 2 } } { 12 ^ { 2 } }$$

Solve for h

$$h=12\sqrt{2}-12\approx 4.970562748$$

$$h=-12\sqrt{2}-12\approx -28.970562748$$

Steps Using the Quadratic Formula

Anything divided by one gives itself.

$$2=\frac{\left(12+h\right)^{2}}{12^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+h\right)^{2}.

$$2=\frac{144+24h+h^{2}}{12^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$2=\frac{144+24h+h^{2}}{144}$$

Divide each term of 144+24h+h^{2} by 144 to get 1+\frac{1}{6}h+\frac{1}{144}h^{2}.

$$2=1+\frac{1}{6}h+\frac{1}{144}h^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}=2$$

Subtract 2 from both sides.

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}-2=0$$

Subtract 2 from 1 to get -1.

$$-1+\frac{1}{6}h+\frac{1}{144}h^{2}=0$$

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.

$$\frac{1}{144}h^{2}+\frac{1}{6}h-1=0$$

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}.

$$h=\frac{-\frac{1}{6}±\sqrt{\left(\frac{1}{6}\right)^{2}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}$$

Multiply -4 times \frac{1}{144}.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-\frac{1}{36}\left(-1\right)}}{2\times \frac{1}{144}}$$

Multiply -\frac{1}{36} times -1.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1+1}{36}}}{2\times \frac{1}{144}}$$

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.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{18}}}{2\times \frac{1}{144}}$$

Take the square root of \frac{1}{18}.

$$h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{2\times \frac{1}{144}}$$

Multiply 2 times \frac{1}{144}.

$$h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}}$$

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}.

$$h=\frac{\sqrt{2}-1}{\frac{1}{72}\times 6}$$

Divide \frac{-1+\sqrt{2}}{6} by \frac{1}{72} by multiplying \frac{-1+\sqrt{2}}{6} by the reciprocal of \frac{1}{72}.

$$h=12\sqrt{2}-12$$

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}.

$$h=\frac{-\sqrt{2}-1}{\frac{1}{72}\times 6}$$

Divide \frac{-1-\sqrt{2}}{6} by \frac{1}{72} by multiplying \frac{-1-\sqrt{2}}{6} by the reciprocal of \frac{1}{72}.

$$h=-12\sqrt{2}-12$$

The equation is now solved.

$$h=12\sqrt{2}-12 h=-12\sqrt{2}-12$$

Steps for Completing the Square

Anything divided by one gives itself.

$$2=\frac{\left(12+h\right)^{2}}{12^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+h\right)^{2}.

$$2=\frac{144+24h+h^{2}}{12^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$2=\frac{144+24h+h^{2}}{144}$$

Divide each term of 144+24h+h^{2} by 144 to get 1+\frac{1}{6}h+\frac{1}{144}h^{2}.

$$2=1+\frac{1}{6}h+\frac{1}{144}h^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}=2$$

Subtract 1 from both sides.

$$\frac{1}{6}h+\frac{1}{144}h^{2}=2-1$$

Subtract 1 from 2 to get 1.

$$\frac{1}{6}h+\frac{1}{144}h^{2}=1$$

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.

$$\frac{1}{144}h^{2}+\frac{1}{6}h=1$$

Multiply both sides by 144.

$$\frac{\frac{1}{144}h^{2}+\frac{1}{6}h}{\frac{1}{144}}=\frac{1}{\frac{1}{144}}$$

Dividing by \frac{1}{144} undoes the multiplication by \frac{1}{144}.

$$h^{2}+\frac{\frac{1}{6}}{\frac{1}{144}}h=\frac{1}{\frac{1}{144}}$$

Divide \frac{1}{6} by \frac{1}{144} by multiplying \frac{1}{6} by the reciprocal of \frac{1}{144}.

$$h^{2}+24h=\frac{1}{\frac{1}{144}}$$

Divide 1 by \frac{1}{144} by multiplying 1 by the reciprocal of \frac{1}{144}.

$$h^{2}+24h=144$$

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.

$$h^{2}+24h+12^{2}=144+12^{2}$$

Square 12.

$$h^{2}+24h+144=144+144$$

Add 144 to 144.

$$h^{2}+24h+144=288$$

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}.

$$\left(h+12\right)^{2}=288$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+12\right)^{2}}=\sqrt{288}$$

Simplify.

$$h+12=12\sqrt{2} h+12=-12\sqrt{2}$$

Subtract 12 from both sides of the equation.

$$h=12\sqrt{2}-12 h=-12\sqrt{2}-12$$