Solve 7^2+8^2-x^2/2*7*8=1/7

$$\frac { 7 ^ { 2 } + 8 ^ { 2 } - x ^ { 2 } } { 2 \times 7 \times 8 } = \frac { 1 } { 7 }$$

Solve for x

$$x=\sqrt{97}\approx 9.848857802$$

$$x=-\sqrt{97}\approx -9.848857802$$

Steps by Finding Square Root

Calculate 7 to the power of 2 and get 49.

$$\frac{49+8^{2}-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{49+64-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Add 49 and 64 to get 113.

$$\frac{113-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Multiply 2 and 7 to get 14.

$$\frac{113-x^{2}}{14\times 8}=\frac{1}{7}$$

Multiply 14 and 8 to get 112.

$$\frac{113-x^{2}}{112}=\frac{1}{7}$$

Divide each term of 113-x^{2} by 112 to get \frac{113}{112}-\frac{1}{112}x^{2}.

$$\frac{113}{112}-\frac{1}{112}x^{2}=\frac{1}{7}$$

Subtract \frac{113}{112} from both sides.

$$-\frac{1}{112}x^{2}=\frac{1}{7}-\frac{113}{112}$$

Subtract \frac{113}{112} from \frac{1}{7} to get -\frac{97}{112}.

$$-\frac{1}{112}x^{2}=-\frac{97}{112}$$

Multiply both sides by -112, the reciprocal of -\frac{1}{112}.

$$x^{2}=-\frac{97}{112}\left(-112\right)$$

Multiply -\frac{97}{112} and -112 to get 97.

$$x^{2}=97$$

Take the square root of both sides of the equation.

$$x=\sqrt{97} x=-\sqrt{97}$$

Steps Using the Quadratic Formula

Calculate 7 to the power of 2 and get 49.

$$\frac{49+8^{2}-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{49+64-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Add 49 and 64 to get 113.

$$\frac{113-x^{2}}{2\times 7\times 8}=\frac{1}{7}$$

Multiply 2 and 7 to get 14.

$$\frac{113-x^{2}}{14\times 8}=\frac{1}{7}$$

Multiply 14 and 8 to get 112.

$$\frac{113-x^{2}}{112}=\frac{1}{7}$$

Divide each term of 113-x^{2} by 112 to get \frac{113}{112}-\frac{1}{112}x^{2}.

$$\frac{113}{112}-\frac{1}{112}x^{2}=\frac{1}{7}$$

Subtract \frac{1}{7} from both sides.

$$\frac{113}{112}-\frac{1}{112}x^{2}-\frac{1}{7}=0$$

Subtract \frac{1}{7} from \frac{113}{112} to get \frac{97}{112}.

$$\frac{97}{112}-\frac{1}{112}x^{2}=0$$

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.

$$-\frac{1}{112}x^{2}+\frac{97}{112}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{112} for a, 0 for b, and \frac{97}{112} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{1}{112}\right)\times \frac{97}{112}}}{2\left(-\frac{1}{112}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{1}{112}\right)\times \frac{97}{112}}}{2\left(-\frac{1}{112}\right)}$$

Multiply -4 times -\frac{1}{112}.

$$x=\frac{0±\sqrt{\frac{1}{28}\times \frac{97}{112}}}{2\left(-\frac{1}{112}\right)}$$

Multiply \frac{1}{28} times \frac{97}{112} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0±\sqrt{\frac{97}{3136}}}{2\left(-\frac{1}{112}\right)}$$

Take the square root of \frac{97}{3136}.

$$x=\frac{0±\frac{\sqrt{97}}{56}}{2\left(-\frac{1}{112}\right)}$$

Multiply 2 times -\frac{1}{112}.

$$x=\frac{0±\frac{\sqrt{97}}{56}}{-\frac{1}{56}}$$

Now solve the equation x=\frac{0±\frac{\sqrt{97}}{56}}{-\frac{1}{56}} when ± is plus.

$$x=-\sqrt{97}$$

Now solve the equation x=\frac{0±\frac{\sqrt{97}}{56}}{-\frac{1}{56}} when ± is minus.

$$x=\sqrt{97}$$

The equation is now solved.

$$x=-\sqrt{97} x=\sqrt{97}$$