Solve sqrt{2x+14}-sqrt{x-7}=sqrt{x+5}

$$\sqrt { 2 x + 14 } - \sqrt { x - 7 } = \sqrt { x + 5 }$$

Solve for x

$$x=11$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2x+14}-\sqrt{x-7}\right)^{2}=\left(\sqrt{x+5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2x+14}-\sqrt{x-7}\right)^{2}.

$$\left(\sqrt{2x+14}\right)^{2}-2\sqrt{2x+14}\sqrt{x-7}+\left(\sqrt{x-7}\right)^{2}=\left(\sqrt{x+5}\right)^{2}$$

Calculate \sqrt{2x+14} to the power of 2 and get 2x+14.

$$2x+14-2\sqrt{2x+14}\sqrt{x-7}+\left(\sqrt{x-7}\right)^{2}=\left(\sqrt{x+5}\right)^{2}$$

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

$$2x+14-2\sqrt{2x+14}\sqrt{x-7}+x-7=\left(\sqrt{x+5}\right)^{2}$$

Combine 2x and x to get 3x.

$$3x+14-2\sqrt{2x+14}\sqrt{x-7}-7=\left(\sqrt{x+5}\right)^{2}$$

Subtract 7 from 14 to get 7.

$$3x+7-2\sqrt{2x+14}\sqrt{x-7}=\left(\sqrt{x+5}\right)^{2}$$

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

$$3x+7-2\sqrt{2x+14}\sqrt{x-7}=x+5$$

Subtract 3x+7 from both sides of the equation.

$$-2\sqrt{2x+14}\sqrt{x-7}=x+5-\left(3x+7\right)$$

To find the opposite of 3x+7, find the opposite of each term.

$$-2\sqrt{2x+14}\sqrt{x-7}=x+5-3x-7$$

Combine x and -3x to get -2x.

$$-2\sqrt{2x+14}\sqrt{x-7}=-2x+5-7$$

Subtract 7 from 5 to get -2.

$$-2\sqrt{2x+14}\sqrt{x-7}=-2x-2$$

Square both sides of the equation.

$$\left(-2\sqrt{2x+14}\sqrt{x-7}\right)^{2}=\left(-2x-2\right)^{2}$$

Expand \left(-2\sqrt{2x+14}\sqrt{x-7}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{2x+14}\right)^{2}\left(\sqrt{x-7}\right)^{2}=\left(-2x-2\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{2x+14}\right)^{2}\left(\sqrt{x-7}\right)^{2}=\left(-2x-2\right)^{2}$$

Calculate \sqrt{2x+14} to the power of 2 and get 2x+14.

$$4\left(2x+14\right)\left(\sqrt{x-7}\right)^{2}=\left(-2x-2\right)^{2}$$

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

$$4\left(2x+14\right)\left(x-7\right)=\left(-2x-2\right)^{2}$$

Use the distributive property to multiply 4 by 2x+14.

$$\left(8x+56\right)\left(x-7\right)=\left(-2x-2\right)^{2}$$

Apply the distributive property by multiplying each term of 8x+56 by each term of x-7.

$$8x^{2}-56x+56x-392=\left(-2x-2\right)^{2}$$

Combine -56x and 56x to get 0.

$$8x^{2}-392=\left(-2x-2\right)^{2}$$

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

$$8x^{2}-392=4x^{2}+8x+4$$

Subtract 4x^{2} from both sides.

$$8x^{2}-392-4x^{2}=8x+4$$

Combine 8x^{2} and -4x^{2} to get 4x^{2}.

$$4x^{2}-392=8x+4$$

Subtract 8x from both sides.

$$4x^{2}-392-8x=4$$

Subtract 4 from both sides.

$$4x^{2}-392-8x-4=0$$

Subtract 4 from -392 to get -396.

$$4x^{2}-396-8x=0$$

Divide both sides by 4.

$$x^{2}-99-2x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-2x-99=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-99. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-99\right)=-99$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -99.

$$1,-99 3,-33 9,-11$$

Calculate the sum for each pair.

$$1-99=-98 3-33=-30 9-11=-2$$

The solution is the pair that gives sum -2.

$$a=-11 b=9$$

Rewrite x^{2}-2x-99 as \left(x^{2}-11x\right)+\left(9x-99\right).

$$\left(x^{2}-11x\right)+\left(9x-99\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-11\right)+9\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x+9\right)$$

To find equation solutions, solve x-11=0 and x+9=0.

$$x=11 x=-9$$

Substitute -9 for x in the equation \sqrt{2x+14}-\sqrt{x-7}=\sqrt{x+5}. The expression \sqrt{2\left(-9\right)+14} is undefined because the radicand cannot be negative.

$$\sqrt{2\left(-9\right)+14}-\sqrt{-9-7}=\sqrt{-9+5}$$

Substitute 11 for x in the equation \sqrt{2x+14}-\sqrt{x-7}=\sqrt{x+5}.

$$\sqrt{2\times 11+14}-\sqrt{11-7}=\sqrt{11+5}$$

Simplify. The value x=11 satisfies the equation.

$$4=4$$

Equation \sqrt{2x+14}-\sqrt{x-7}=\sqrt{x+5} has a unique solution.

$$x=11$$