Solve sqrt{x+15}+sqrt{x+7}=4/sqrt{x+7}

$$\sqrt { x + 15 } + \sqrt { x + 7 } = \frac { 4 } { \sqrt { x + 7 } }$$

Solve for x

$$x=-6$$

Solution Steps

Subtract \sqrt{x+7} from both sides of the equation.

$$\sqrt{x+15}=\frac{4}{\sqrt{x+7}}-\sqrt{x+7}$$

Square both sides of the equation.

$$\left(\sqrt{x+15}\right)^{2}=\left(\frac{4}{\sqrt{x+7}}-\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{x+15} to the power of 2 and get x+15.

$$x+15=\left(\frac{4}{\sqrt{x+7}}-\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{4}{\sqrt{x+7}}-\sqrt{x+7}\right)^{2}.

$$x+15=\left(\frac{4}{\sqrt{x+7}}\right)^{2}-2\times \frac{4}{\sqrt{x+7}}\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

To raise \frac{4}{\sqrt{x+7}} to a power, raise both numerator and denominator to the power and then divide.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}-2\times \frac{4}{\sqrt{x+7}}\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

Express -2\times \frac{4}{\sqrt{x+7}} as a single fraction.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4}{\sqrt{x+7}}\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

Express \frac{-2\times 4}{\sqrt{x+7}}\sqrt{x+7} as a single fraction.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}+\left(\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}+x+7$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+7 times \frac{\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}+\frac{\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}$$

Since \frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}} and \frac{\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}} have the same denominator, add them by adding their numerators.

$$x+15=\frac{4^{2}+\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+7 times \frac{\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}+\frac{\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}$$

Since \frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}} and \frac{\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}} have the same denominator, add them by adding their numerators.

$$x+15=\frac{4^{2}+\left(x+7\right)\left(\sqrt{x+7}\right)^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+7 times \frac{\sqrt{x+7}}{\sqrt{x+7}}.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}}+\frac{\left(x+7\right)\sqrt{x+7}}{\sqrt{x+7}}$$

Since \frac{-2\times 4\sqrt{x+7}}{\sqrt{x+7}} and \frac{\left(x+7\right)\sqrt{x+7}}{\sqrt{x+7}} have the same denominator, add them by adding their numerators.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-2\times 4\sqrt{x+7}+\left(x+7\right)\sqrt{x+7}}{\sqrt{x+7}}$$

Do the multiplications in -2\times 4\sqrt{x+7}+\left(x+7\right)\sqrt{x+7}.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-8\sqrt{x+7}+x\sqrt{x+7}+7\sqrt{x+7}}{\sqrt{x+7}}$$

Combine like terms in -8\sqrt{x+7}+x\sqrt{x+7}+7\sqrt{x+7}.

$$x+15=\frac{4^{2}}{\left(\sqrt{x+7}\right)^{2}}+\frac{-\sqrt{x+7}+x\sqrt{x+7}}{\sqrt{x+7}}$$

Calculate 4 to the power of 2 and get 16.

$$x+15=\frac{16}{\left(\sqrt{x+7}\right)^{2}}+\frac{-\sqrt{x+7}+x\sqrt{x+7}}{\sqrt{x+7}}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x+15=\frac{16}{x+7}+\frac{-\sqrt{x+7}+x\sqrt{x+7}}{\sqrt{x+7}}$$

Multiply both sides of the equation by x+7.

$$\left(x+7\right)x+\left(x+7\right)\times 15=16+\left(x+7\right)^{\frac{1}{2}}\left(-\sqrt{x+7}+x\sqrt{x+7}\right)$$

Reorder the terms.

$$x\left(x+7\right)+15\left(x+7\right)=\sqrt{x+7}\left(\sqrt{x+7}x-\sqrt{x+7}\right)+16$$

Use the distributive property to multiply x by x+7.

$$x^{2}+7x+15\left(x+7\right)=\sqrt{x+7}\left(\sqrt{x+7}x-\sqrt{x+7}\right)+16$$

Use the distributive property to multiply 15 by x+7.

$$x^{2}+7x+15x+105=\sqrt{x+7}\left(\sqrt{x+7}x-\sqrt{x+7}\right)+16$$

Combine 7x and 15x to get 22x.

$$x^{2}+22x+105=\sqrt{x+7}\left(\sqrt{x+7}x-\sqrt{x+7}\right)+16$$

Use the distributive property to multiply \sqrt{x+7} by \sqrt{x+7}x-\sqrt{x+7}.

$$x^{2}+22x+105=x\left(\sqrt{x+7}\right)^{2}-\left(\sqrt{x+7}\right)^{2}+16$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x^{2}+22x+105=x\left(x+7\right)-\left(\sqrt{x+7}\right)^{2}+16$$

Use the distributive property to multiply x by x+7.

$$x^{2}+22x+105=x^{2}+7x-\left(\sqrt{x+7}\right)^{2}+16$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x^{2}+22x+105=x^{2}+7x-\left(x+7\right)+16$$

To find the opposite of x+7, find the opposite of each term.

$$x^{2}+22x+105=x^{2}+7x-x-7+16$$

Combine 7x and -x to get 6x.

$$x^{2}+22x+105=x^{2}+6x-7+16$$

Add -7 and 16 to get 9.

$$x^{2}+22x+105=x^{2}+6x+9$$

Subtract x^{2} from both sides.

$$x^{2}+22x+105-x^{2}=6x+9$$

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

$$22x+105=6x+9$$

Subtract 6x from both sides.

$$22x+105-6x=9$$

Combine 22x and -6x to get 16x.

$$16x+105=9$$

Subtract 105 from both sides.

$$16x=9-105$$

Subtract 105 from 9 to get -96.

$$16x=-96$$

Divide both sides by 16.

$$x=\frac{-96}{16}$$

Divide -96 by 16 to get -6.

$$x=-6$$

Substitute -6 for x in the equation \sqrt{x+15}+\sqrt{x+7}=\frac{4}{\sqrt{x+7}}.

$$\sqrt{-6+15}+\sqrt{-6+7}=\frac{4}{\sqrt{-6+7}}$$

Simplify. The value x=-6 satisfies the equation.

$$4=4$$

Equation \sqrt{x+15}=-\sqrt{x+7}+\frac{4}{\sqrt{x+7}} has a unique solution.

$$x=-6$$