Solve sqrt{x+7}-sqrt{x-5}=2%

$$\sqrt{ x+7 } - \sqrt{ x-5 } = 2 \%$$

Solve for x

$$x = \frac{899990001}{10000} = 89999\frac{1}{10000} = 89999.0001$$

Solution Steps

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

$$\sqrt{x+7}=\frac{2}{100}+\sqrt{x-5}$$

Reduce the fraction \frac{2}{100} to lowest terms by extracting and canceling out 2.

$$\sqrt{x+7}=\frac{1}{50}+\sqrt{x-5}$$

Square both sides of the equation.

$$\left(\sqrt{x+7}\right)^{2}=\left(\frac{1}{50}+\sqrt{x-5}\right)^{2}$$

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

$$x+7=\left(\frac{1}{50}+\sqrt{x-5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{1}{50}+\sqrt{x-5}\right)^{2}.

$$x+7=\frac{1}{2500}+\frac{1}{25}\sqrt{x-5}+\left(\sqrt{x-5}\right)^{2}$$

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

$$x+7=\frac{1}{2500}+\frac{1}{25}\sqrt{x-5}+x-5$$

Convert 5 to fraction \frac{12500}{2500}.

$$x+7=\frac{1}{2500}+\frac{1}{25}\sqrt{x-5}+x-\frac{12500}{2500}$$

Since \frac{1}{2500} and \frac{12500}{2500} have the same denominator, subtract them by subtracting their numerators.

$$x+7=\frac{1-12500}{2500}+\frac{1}{25}\sqrt{x-5}+x$$

Subtract 12500 from 1 to get -12499.

$$x+7=-\frac{12499}{2500}+\frac{1}{25}\sqrt{x-5}+x$$

Subtract \frac{1}{25}\sqrt{x-5} from both sides.

$$x+7-\frac{1}{25}\sqrt{x-5}=-\frac{12499}{2500}+x$$

Subtract x from both sides.

$$x+7-\frac{1}{25}\sqrt{x-5}-x=-\frac{12499}{2500}$$

Combine x and -x to get 0.

$$7-\frac{1}{25}\sqrt{x-5}=-\frac{12499}{2500}$$

Subtract 7 from both sides.

$$-\frac{1}{25}\sqrt{x-5}=-\frac{12499}{2500}-7$$

Convert 7 to fraction \frac{17500}{2500}.

$$-\frac{1}{25}\sqrt{x-5}=-\frac{12499}{2500}-\frac{17500}{2500}$$

Since -\frac{12499}{2500} and \frac{17500}{2500} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{25}\sqrt{x-5}=\frac{-12499-17500}{2500}$$

Subtract 17500 from -12499 to get -29999.

$$-\frac{1}{25}\sqrt{x-5}=-\frac{29999}{2500}$$

Multiply both sides by -25, the reciprocal of -\frac{1}{25}.

$$\sqrt{x-5}=-\frac{29999}{2500}\left(-25\right)$$

Express -\frac{29999}{2500}\left(-25\right) as a single fraction.

$$\sqrt{x-5}=\frac{-29999\left(-25\right)}{2500}$$

Multiply -29999 and -25 to get 749975.

$$\sqrt{x-5}=\frac{749975}{2500}$$

Reduce the fraction \frac{749975}{2500} to lowest terms by extracting and canceling out 25.

$$\sqrt{x-5}=\frac{29999}{100}$$

Square both sides of the equation.

$$x-5=\frac{899940001}{10000}$$

Add 5 to both sides of the equation.

$$x-5-\left(-5\right)=\frac{899940001}{10000}-\left(-5\right)$$

Subtracting -5 from itself leaves 0.

$$x=\frac{899940001}{10000}-\left(-5\right)$$

Subtract -5 from \frac{899940001}{10000}.

$$x=\frac{899990001}{10000}$$

Substitute \frac{899990001}{10000} for x in the equation \sqrt{x+7}-\sqrt{x-5}=\frac{2}{100}.

$$\sqrt{\frac{899990001}{10000}+7}-\sqrt{\frac{899990001}{10000}-5}=\frac{2}{100}$$

Simplify. The value x=\frac{899990001}{10000} satisfies the equation.

$$\frac{1}{50}=\frac{1}{50}$$

Equation \sqrt{x+7}=\sqrt{x-5}+\frac{1}{50} has a unique solution.

$$x=\frac{899990001}{10000}$$