Solve sqrt{x}+sqrt{x+7}=17

$$\sqrt { x } + \sqrt { x + 7 } = 17$$

Solve for x

$$x = \frac{19881}{289} = 68\frac{229}{289} \approx 68.792387543$$

Solution Steps

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

$$\sqrt{x}=17-\sqrt{x+7}$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(17-\sqrt{x+7}\right)^{2}$$

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

$$x=\left(17-\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(17-\sqrt{x+7}\right)^{2}.

$$x=289-34\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

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

$$x=289-34\sqrt{x+7}+x+7$$

Add 289 and 7 to get 296.

$$x=296-34\sqrt{x+7}+x$$

Add 34\sqrt{x+7} to both sides.

$$x+34\sqrt{x+7}=296+x$$

Subtract x from both sides.

$$x+34\sqrt{x+7}-x=296$$

Combine x and -x to get 0.

$$34\sqrt{x+7}=296$$

Divide both sides by 34.

$$\sqrt{x+7}=\frac{296}{34}$$

Reduce the fraction \frac{296}{34} to lowest terms by extracting and canceling out 2.

$$\sqrt{x+7}=\frac{148}{17}$$

Square both sides of the equation.

$$x+7=\frac{21904}{289}$$

Subtract 7 from both sides of the equation.

$$x+7-7=\frac{21904}{289}-7$$

Subtracting 7 from itself leaves 0.

$$x=\frac{21904}{289}-7$$

Subtract 7 from \frac{21904}{289}.

$$x=\frac{19881}{289}$$

Substitute \frac{19881}{289} for x in the equation \sqrt{x}+\sqrt{x+7}=17.

$$\sqrt{\frac{19881}{289}}+\sqrt{\frac{19881}{289}+7}=17$$

Simplify. The value x=\frac{19881}{289} satisfies the equation.

$$17=17$$

Equation \sqrt{x}=-\sqrt{x+7}+17 has a unique solution.

$$x=\frac{19881}{289}$$