Solve sqrt{2x+7}-sqrt{x+7}=-1

$$\sqrt { 2 x + 7 } - \sqrt { x + 7 } = - 1$$

Solve for x

$$x=-3$$

Solution Steps

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

$$\sqrt{2x+7}=-1+\sqrt{x+7}$$

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1+\sqrt{x+7}\right)^{2}.

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

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

$$2x+7=1-2\sqrt{x+7}+x+7$$

Add 1 and 7 to get 8.

$$2x+7=8-2\sqrt{x+7}+x$$

Subtract 8+x from both sides of the equation.

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

To find the opposite of 8+x, find the opposite of each term.

$$2x+7-8-x=-2\sqrt{x+7}$$

Subtract 8 from 7 to get -1.

$$2x-1-x=-2\sqrt{x+7}$$

Combine 2x and -x to get x.

$$x-1=-2\sqrt{x+7}$$

Square both sides of the equation.

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

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

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

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

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

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

$$x^{2}-2x+1=4\left(\sqrt{x+7}\right)^{2}$$

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

$$x^{2}-2x+1=4\left(x+7\right)$$

Use the distributive property to multiply 4 by x+7.

$$x^{2}-2x+1=4x+28$$

Subtract 4x from both sides.

$$x^{2}-2x+1-4x=28$$

Combine -2x and -4x to get -6x.

$$x^{2}-6x+1=28$$

Subtract 28 from both sides.

$$x^{2}-6x+1-28=0$$

Subtract 28 from 1 to get -27.

$$x^{2}-6x-27=0$$

To solve the equation, factor x^{2}-6x-27 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.

$$a+b=-6 ab=-27$$

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 -27.

$$1,-27 3,-9$$

Calculate the sum for each pair.

$$1-27=-26 3-9=-6$$

The solution is the pair that gives sum -6.

$$a=-9 b=3$$

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

$$\left(x-9\right)\left(x+3\right)$$

To find equation solutions, solve x-9=0 and x+3=0.

$$x=9 x=-3$$

Substitute 9 for x in the equation \sqrt{2x+7}-\sqrt{x+7}=-1.

$$\sqrt{2\times 9+7}-\sqrt{9+7}=-1$$

Simplify. The value x=9 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Substitute -3 for x in the equation \sqrt{2x+7}-\sqrt{x+7}=-1.

$$\sqrt{2\left(-3\right)+7}-\sqrt{-3+7}=-1$$

Simplify. The value x=-3 satisfies the equation.

$$-1=-1$$

Equation \sqrt{2x+7}=\sqrt{x+7}-1 has a unique solution.

$$x=-3$$