Solve sqrt{x}+sqrt{x+13}=91/sqrt{x+13}

$$\sqrt { x } + \sqrt { x + 13 } = \frac { 91 } { \sqrt { x + 13 } }$$

Solve for x

$$x=36$$

Solution Steps

Subtract \sqrt{x+13} from both sides of the equation.

$$\sqrt{x}=\frac{91}{\sqrt{x+13}}-\sqrt{x+13}$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(\frac{91}{\sqrt{x+13}}-\sqrt{x+13}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$x=\left(\frac{91}{\sqrt{x+13}}-\sqrt{x+13}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{91}{\sqrt{x+13}}-\sqrt{x+13}\right)^{2}.

$$x=\left(\frac{91}{\sqrt{x+13}}\right)^{2}-2\times \frac{91}{\sqrt{x+13}}\sqrt{x+13}+\left(\sqrt{x+13}\right)^{2}$$

To raise \frac{91}{\sqrt{x+13}} to a power, raise both numerator and denominator to the power and then divide.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}-2\times \frac{91}{\sqrt{x+13}}\sqrt{x+13}+\left(\sqrt{x+13}\right)^{2}$$

Express -2\times \frac{91}{\sqrt{x+13}} as a single fraction.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91}{\sqrt{x+13}}\sqrt{x+13}+\left(\sqrt{x+13}\right)^{2}$$

Express \frac{-2\times 91}{\sqrt{x+13}}\sqrt{x+13} as a single fraction.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}+\left(\sqrt{x+13}\right)^{2}$$

Calculate \sqrt{x+13} to the power of 2 and get x+13.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}+x+13$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+13 times \frac{\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}+\frac{\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}$$

Since \frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}} and \frac{\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}} have the same denominator, add them by adding their numerators.

$$x=\frac{91^{2}+\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+13 times \frac{\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}+\frac{\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}$$

Since \frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}} and \frac{\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}} have the same denominator, add them by adding their numerators.

$$x=\frac{91^{2}+\left(x+13\right)\left(\sqrt{x+13}\right)^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+13 times \frac{\sqrt{x+13}}{\sqrt{x+13}}.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}}+\frac{\left(x+13\right)\sqrt{x+13}}{\sqrt{x+13}}$$

Since \frac{-2\times 91\sqrt{x+13}}{\sqrt{x+13}} and \frac{\left(x+13\right)\sqrt{x+13}}{\sqrt{x+13}} have the same denominator, add them by adding their numerators.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-2\times 91\sqrt{x+13}+\left(x+13\right)\sqrt{x+13}}{\sqrt{x+13}}$$

Do the multiplications in -2\times 91\sqrt{x+13}+\left(x+13\right)\sqrt{x+13}.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-182\sqrt{x+13}+x\sqrt{x+13}+13\sqrt{x+13}}{\sqrt{x+13}}$$

Combine like terms in -182\sqrt{x+13}+x\sqrt{x+13}+13\sqrt{x+13}.

$$x=\frac{91^{2}}{\left(\sqrt{x+13}\right)^{2}}+\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Calculate 91 to the power of 2 and get 8281.

$$x=\frac{8281}{\left(\sqrt{x+13}\right)^{2}}+\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Calculate \sqrt{x+13} to the power of 2 and get x+13.

$$x=\frac{8281}{x+13}+\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Subtract \frac{8281}{x+13} from both sides.

$$x-\frac{8281}{x+13}=\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+13}{x+13}.

$$\frac{x\left(x+13\right)}{x+13}-\frac{8281}{x+13}=\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Since \frac{x\left(x+13\right)}{x+13} and \frac{8281}{x+13} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+13\right)-8281}{x+13}=\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Do the multiplications in x\left(x+13\right)-8281.

$$\frac{x^{2}+13x-8281}{x+13}=\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}$$

Subtract \frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}} from both sides.

$$\frac{x^{2}+13x-8281}{x+13}-\frac{-169\sqrt{x+13}+x\sqrt{x+13}}{\sqrt{x+13}}=0$$

Variable x cannot be equal to -13 since division by zero is not defined. Multiply both sides of the equation by x+13.

$$x^{2}+13x-8281-\left(x+13\right)^{\frac{1}{2}}\left(-169\sqrt{x+13}+x\sqrt{x+13}\right)=0$$

Reorder the terms.

$$-\sqrt{x+13}\left(\sqrt{x+13}x-169\sqrt{x+13}\right)+x^{2}+13x-8281=0$$

Use the distributive property to multiply -\sqrt{x+13} by \sqrt{x+13}x-169\sqrt{x+13}.

$$-x\left(\sqrt{x+13}\right)^{2}+169\left(\sqrt{x+13}\right)^{2}+x^{2}+13x-8281=0$$

Calculate \sqrt{x+13} to the power of 2 and get x+13.

$$-x\left(x+13\right)+169\left(\sqrt{x+13}\right)^{2}+x^{2}+13x-8281=0$$

Use the distributive property to multiply -x by x+13.

$$-x^{2}-13x+169\left(\sqrt{x+13}\right)^{2}+x^{2}+13x-8281=0$$

Calculate \sqrt{x+13} to the power of 2 and get x+13.

$$-x^{2}-13x+169\left(x+13\right)+x^{2}+13x-8281=0$$

Use the distributive property to multiply 169 by x+13.

$$-x^{2}-13x+169x+2197+x^{2}+13x-8281=0$$

Combine -13x and 169x to get 156x.

$$-x^{2}+156x+2197+x^{2}+13x-8281=0$$

Combine -x^{2} and x^{2} to get 0.

$$156x+2197+13x-8281=0$$

Combine 156x and 13x to get 169x.

$$169x+2197-8281=0$$

Subtract 8281 from 2197 to get -6084.

$$169x-6084=0$$

Add 6084 to both sides. Anything plus zero gives itself.

$$169x=6084$$

Divide both sides by 169.

$$x=\frac{6084}{169}$$

Divide 6084 by 169 to get 36.

$$x=36$$

Substitute 36 for x in the equation \sqrt{x}+\sqrt{x+13}=\frac{91}{\sqrt{x+13}}.

$$\sqrt{36}+\sqrt{36+13}=\frac{91}{\sqrt{36+13}}$$

Simplify. The value x=36 satisfies the equation.

$$13=13$$

Equation \sqrt{x}=-\sqrt{x+13}+\frac{91}{\sqrt{x+13}} has a unique solution.

$$x=36$$