Solve sqrt{x+14}-sqrt{2x+5}=1

$$\sqrt { x + 14 } - \sqrt { 2 x + 5 } = 1$$

Solve for x

$$x=2$$

Solution Steps

Subtract -\sqrt{2x+5} from both sides of the equation.

$$\sqrt{x+14}=1+\sqrt{2x+5}$$

Square both sides of the equation.

$$\left(\sqrt{x+14}\right)^{2}=\left(1+\sqrt{2x+5}\right)^{2}$$

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

$$x+14=\left(1+\sqrt{2x+5}\right)^{2}$$

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

$$x+14=1+2\sqrt{2x+5}+\left(\sqrt{2x+5}\right)^{2}$$

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

$$x+14=1+2\sqrt{2x+5}+2x+5$$

Add 1 and 5 to get 6.

$$x+14=6+2\sqrt{2x+5}+2x$$

Subtract 6+2x from both sides of the equation.

$$x+14-\left(6+2x\right)=2\sqrt{2x+5}$$

To find the opposite of 6+2x, find the opposite of each term.

$$x+14-6-2x=2\sqrt{2x+5}$$

Subtract 6 from 14 to get 8.

$$x+8-2x=2\sqrt{2x+5}$$

Combine x and -2x to get -x.

$$-x+8=2\sqrt{2x+5}$$

Square both sides of the equation.

$$\left(-x+8\right)^{2}=\left(2\sqrt{2x+5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x+8\right)^{2}.

$$x^{2}-16x+64=\left(2\sqrt{2x+5}\right)^{2}$$

Expand \left(2\sqrt{2x+5}\right)^{2}.

$$x^{2}-16x+64=2^{2}\left(\sqrt{2x+5}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-16x+64=4\left(\sqrt{2x+5}\right)^{2}$$

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

$$x^{2}-16x+64=4\left(2x+5\right)$$

Use the distributive property to multiply 4 by 2x+5.

$$x^{2}-16x+64=8x+20$$

Subtract 8x from both sides.

$$x^{2}-16x+64-8x=20$$

Combine -16x and -8x to get -24x.

$$x^{2}-24x+64=20$$

Subtract 20 from both sides.

$$x^{2}-24x+64-20=0$$

Subtract 20 from 64 to get 44.

$$x^{2}-24x+44=0$$

To solve the equation, factor x^{2}-24x+44 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-24 ab=44$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 44.

$$-1,-44 -2,-22 -4,-11$$

Calculate the sum for each pair.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

The solution is the pair that gives sum -24.

$$a=-22 b=-2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-22\right)\left(x-2\right)$$

To find equation solutions, solve x-22=0 and x-2=0.

$$x=22 x=2$$

Substitute 22 for x in the equation \sqrt{x+14}-\sqrt{2x+5}=1.

$$\sqrt{22+14}-\sqrt{2\times 22+5}=1$$

Simplify. The value x=22 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute 2 for x in the equation \sqrt{x+14}-\sqrt{2x+5}=1.

$$\sqrt{2+14}-\sqrt{2\times 2+5}=1$$

Simplify. The value x=2 satisfies the equation.

$$1=1$$

Equation \sqrt{x+14}=\sqrt{2x+5}+1 has a unique solution.

$$x=2$$