Solve sqrt{5-x}+sqrt{5+x}=12/sqrt{5+x}

$$\sqrt { 5 - x } + \sqrt { 5 + x } = \frac { 12 } { \sqrt { 5 + x } }$$

Solve for x

$$x=4$$

$$x=3$$

Solution Steps

Subtract \sqrt{5+x} from both sides of the equation.

$$\sqrt{5-x}=\frac{12}{\sqrt{5+x}}-\sqrt{5+x}$$

Square both sides of the equation.

$$\left(\sqrt{5-x}\right)^{2}=\left(\frac{12}{\sqrt{5+x}}-\sqrt{5+x}\right)^{2}$$

Calculate \sqrt{5-x} to the power of 2 and get 5-x.

$$5-x=\left(\frac{12}{\sqrt{5+x}}-\sqrt{5+x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{12}{\sqrt{5+x}}-\sqrt{5+x}\right)^{2}.

$$5-x=\left(\frac{12}{\sqrt{5+x}}\right)^{2}-2\times \frac{12}{\sqrt{5+x}}\sqrt{5+x}+\left(\sqrt{5+x}\right)^{2}$$

To raise \frac{12}{\sqrt{5+x}} to a power, raise both numerator and denominator to the power and then divide.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}-2\times \frac{12}{\sqrt{5+x}}\sqrt{5+x}+\left(\sqrt{5+x}\right)^{2}$$

Express -2\times \frac{12}{\sqrt{5+x}} as a single fraction.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12}{\sqrt{5+x}}\sqrt{5+x}+\left(\sqrt{5+x}\right)^{2}$$

Express \frac{-2\times 12}{\sqrt{5+x}}\sqrt{5+x} as a single fraction.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}+\left(\sqrt{5+x}\right)^{2}$$

Calculate \sqrt{5+x} to the power of 2 and get 5+x.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}+5+x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5+x times \frac{\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}+\frac{\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}$$

Since \frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}} and \frac{\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}} have the same denominator, add them by adding their numerators.

$$5-x=\frac{12^{2}+\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5+x times \frac{\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}+\frac{\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}$$

Since \frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}} and \frac{\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}} have the same denominator, add them by adding their numerators.

$$5-x=\frac{12^{2}+\left(5+x\right)\left(\sqrt{5+x}\right)^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5+x times \frac{\sqrt{5+x}}{\sqrt{5+x}}.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}}+\frac{\left(5+x\right)\sqrt{5+x}}{\sqrt{5+x}}$$

Since \frac{-2\times 12\sqrt{5+x}}{\sqrt{5+x}} and \frac{\left(5+x\right)\sqrt{5+x}}{\sqrt{5+x}} have the same denominator, add them by adding their numerators.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-2\times 12\sqrt{5+x}+\left(5+x\right)\sqrt{5+x}}{\sqrt{5+x}}$$

Do the multiplications in -2\times 12\sqrt{5+x}+\left(5+x\right)\sqrt{5+x}.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-24\sqrt{5+x}+5\sqrt{5+x}+x\sqrt{5+x}}{\sqrt{5+x}}$$

Combine like terms in -24\sqrt{5+x}+5\sqrt{5+x}+x\sqrt{5+x}.

$$5-x=\frac{12^{2}}{\left(\sqrt{5+x}\right)^{2}}+\frac{-19\sqrt{5+x}+x\sqrt{5+x}}{\sqrt{5+x}}$$

Calculate 12 to the power of 2 and get 144.

$$5-x=\frac{144}{\left(\sqrt{5+x}\right)^{2}}+\frac{-19\sqrt{5+x}+x\sqrt{5+x}}{\sqrt{5+x}}$$

Calculate \sqrt{5+x} to the power of 2 and get 5+x.

$$5-x=\frac{144}{5+x}+\frac{-19\sqrt{5+x}+x\sqrt{5+x}}{\sqrt{5+x}}$$

Multiply both sides of the equation by x+5.

$$\left(x+5\right)\times 5-x\left(x+5\right)=144+\left(x+5\right)^{\frac{1}{2}}\left(-19\sqrt{5+x}+x\sqrt{5+x}\right)$$

Reorder the terms.

$$-x\left(x+5\right)+5\left(x+5\right)=\sqrt{x+5}\left(\sqrt{x+5}x-19\sqrt{x+5}\right)+144$$

Use the distributive property to multiply -x by x+5.

$$-x^{2}-5x+5\left(x+5\right)=\sqrt{x+5}\left(\sqrt{x+5}x-19\sqrt{x+5}\right)+144$$

Use the distributive property to multiply 5 by x+5.

$$-x^{2}-5x+5x+25=\sqrt{x+5}\left(\sqrt{x+5}x-19\sqrt{x+5}\right)+144$$

Combine -5x and 5x to get 0.

$$-x^{2}+25=\sqrt{x+5}\left(\sqrt{x+5}x-19\sqrt{x+5}\right)+144$$

Use the distributive property to multiply \sqrt{x+5} by \sqrt{x+5}x-19\sqrt{x+5}.

$$-x^{2}+25=x\left(\sqrt{x+5}\right)^{2}-19\left(\sqrt{x+5}\right)^{2}+144$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$-x^{2}+25=x\left(x+5\right)-19\left(\sqrt{x+5}\right)^{2}+144$$

Use the distributive property to multiply x by x+5.

$$-x^{2}+25=x^{2}+5x-19\left(\sqrt{x+5}\right)^{2}+144$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$-x^{2}+25=x^{2}+5x-19\left(x+5\right)+144$$

Use the distributive property to multiply -19 by x+5.

$$-x^{2}+25=x^{2}+5x-19x-95+144$$

Combine 5x and -19x to get -14x.

$$-x^{2}+25=x^{2}-14x-95+144$$

Add -95 and 144 to get 49.

$$-x^{2}+25=x^{2}-14x+49$$

Subtract x^{2} from both sides.

$$-x^{2}+25-x^{2}=-14x+49$$

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

$$-2x^{2}+25=-14x+49$$

Add 14x to both sides.

$$-2x^{2}+25+14x=49$$

Subtract 49 from both sides.

$$-2x^{2}+25+14x-49=0$$

Subtract 49 from 25 to get -24.

$$-2x^{2}-24+14x=0$$

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.

$$-2x^{2}+14x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 14 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-2\right)\left(-24\right)}}{2\left(-2\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-2\right)\left(-24\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-14±\sqrt{196+8\left(-24\right)}}{2\left(-2\right)}$$

Multiply 8 times -24.

$$x=\frac{-14±\sqrt{196-192}}{2\left(-2\right)}$$

Add 196 to -192.

$$x=\frac{-14±\sqrt{4}}{2\left(-2\right)}$$

Take the square root of 4.

$$x=\frac{-14±2}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-14±2}{-4}$$

Now solve the equation x=\frac{-14±2}{-4} when ± is plus. Add -14 to 2.

$$x=-\frac{12}{-4}$$

Divide -12 by -4.

$$x=3$$

Now solve the equation x=\frac{-14±2}{-4} when ± is minus. Subtract 2 from -14.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

The equation is now solved.

$$x=3 x=4$$

Substitute 3 for x in the equation \sqrt{5-x}+\sqrt{5+x}=\frac{12}{\sqrt{5+x}}.

$$\sqrt{5-3}+\sqrt{5+3}=\frac{12}{\sqrt{5+3}}$$

Simplify. The value x=3 satisfies the equation.

$$3\times 2^{\frac{1}{2}}=3\times 2^{\frac{1}{2}}$$

Substitute 4 for x in the equation \sqrt{5-x}+\sqrt{5+x}=\frac{12}{\sqrt{5+x}}.

$$\sqrt{5-4}+\sqrt{5+4}=\frac{12}{\sqrt{5+4}}$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

List all solutions of \sqrt{5-x}=-\sqrt{x+5}+\frac{12}{\sqrt{x+5}}.

$$x=3 x=4$$