Solve 10^2+20^2-alpha^2/2*10*20=-1/2

$$\frac { 10 ^ { 2 } + 20 ^ { 2 } - \alpha ^ { 2 } } { 2 \times 10 \times 20 } = - \frac { 1 } { 2 }$$

Solve for α

$$\alpha =10\sqrt{7}\approx 26.457513111$$

$$\alpha =-10\sqrt{7}\approx -26.457513111$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 2 and get 100.

$$\frac{100+20^{2}-\alpha ^{2}}{2\times 10\times 20}=-\frac{1}{2}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{100+400-\alpha ^{2}}{2\times 10\times 20}=-\frac{1}{2}$$

Add 100 and 400 to get 500.

$$\frac{500-\alpha ^{2}}{2\times 10\times 20}=-\frac{1}{2}$$

Add \frac{1}{2} to both sides.

$$\frac{500-\alpha ^{2}}{2\times 10\times 20}+\frac{1}{2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 10\times 20 and 2 is 400. Multiply \frac{1}{2} times \frac{200}{200}.

$$\frac{500-\alpha ^{2}}{400}+\frac{200}{400}=0$$

Since \frac{500-\alpha ^{2}}{400} and \frac{200}{400} have the same denominator, add them by adding their numerators.

$$\frac{500-\alpha ^{2}+200}{400}=0$$

Combine like terms in 500-\alpha ^{2}+200.

$$\frac{700-\alpha ^{2}}{400}=0$$

Divide each term of 700-\alpha ^{2} by 400 to get \frac{7}{4}-\frac{1}{400}\alpha ^{2}.

$$\frac{7}{4}-\frac{1}{400}\alpha ^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-\frac{1}{400}\alpha ^{2}+\frac{7}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{400} for a, 0 for b, and \frac{7}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\alpha =\frac{0±\sqrt{0^{2}-4\left(-\frac{1}{400}\right)\times \frac{7}{4}}}{2\left(-\frac{1}{400}\right)}$$

Square 0.

$$\alpha =\frac{0±\sqrt{-4\left(-\frac{1}{400}\right)\times \frac{7}{4}}}{2\left(-\frac{1}{400}\right)}$$

Multiply -4 times -\frac{1}{400}.

$$\alpha =\frac{0±\sqrt{\frac{1}{100}\times \frac{7}{4}}}{2\left(-\frac{1}{400}\right)}$$

Multiply \frac{1}{100} times \frac{7}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\alpha =\frac{0±\sqrt{\frac{7}{400}}}{2\left(-\frac{1}{400}\right)}$$

Take the square root of \frac{7}{400}.

$$\alpha =\frac{0±\frac{\sqrt{7}}{20}}{2\left(-\frac{1}{400}\right)}$$

Multiply 2 times -\frac{1}{400}.

$$\alpha =\frac{0±\frac{\sqrt{7}}{20}}{-\frac{1}{200}}$$

Now solve the equation \alpha =\frac{0±\frac{\sqrt{7}}{20}}{-\frac{1}{200}} when ± is plus.

$$\alpha =-10\sqrt{7}$$

Now solve the equation \alpha =\frac{0±\frac{\sqrt{7}}{20}}{-\frac{1}{200}} when ± is minus.

$$\alpha =10\sqrt{7}$$

The equation is now solved.

$$\alpha =-10\sqrt{7} \alpha =10\sqrt{7}$$