Solve sqrt{x+6}+sqrt{x-29}=7

$$\sqrt{ x+6 } + \sqrt{ x-29 } =7$$

Solve for x

$$x=30$$

Solution Steps

Subtract \sqrt{x-29} from both sides of the equation.

$$\sqrt{x+6}=7-\sqrt{x-29}$$

Square both sides of the equation.

$$\left(\sqrt{x+6}\right)^{2}=\left(7-\sqrt{x-29}\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$x+6=\left(7-\sqrt{x-29}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-\sqrt{x-29}\right)^{2}.

$$x+6=49-14\sqrt{x-29}+\left(\sqrt{x-29}\right)^{2}$$

Calculate \sqrt{x-29} to the power of 2 and get x-29.

$$x+6=49-14\sqrt{x-29}+x-29$$

Subtract 29 from 49 to get 20.

$$x+6=20-14\sqrt{x-29}+x$$

Add 14\sqrt{x-29} to both sides.

$$x+6+14\sqrt{x-29}=20+x$$

Subtract x from both sides.

$$x+6+14\sqrt{x-29}-x=20$$

Combine x and -x to get 0.

$$6+14\sqrt{x-29}=20$$

Subtract 6 from both sides.

$$14\sqrt{x-29}=20-6$$

Subtract 6 from 20 to get 14.

$$14\sqrt{x-29}=14$$

Divide both sides by 14.

$$\sqrt{x-29}=\frac{14}{14}$$

Divide 14 by 14 to get 1.

$$\sqrt{x-29}=1$$

Square both sides of the equation.

$$x-29=1$$

Add 29 to both sides of the equation.

$$x-29-\left(-29\right)=1-\left(-29\right)$$

Subtracting -29 from itself leaves 0.

$$x=1-\left(-29\right)$$

Subtract -29 from 1.

$$x=30$$

Substitute 30 for x in the equation \sqrt{x+6}+\sqrt{x-29}=7.

$$\sqrt{30+6}+\sqrt{30-29}=7$$

Simplify. The value x=30 satisfies the equation.

$$7=7$$

Equation \sqrt{x+6}=-\sqrt{x-29}+7 has a unique solution.

$$x=30$$