Solve sqrt{2x-3}-sqrt{x+7}=4

$$\sqrt { 2 x - 3 } - \sqrt { x + 7 } = 4$$

Solve for x

$$x=114$$

Solution Steps

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

$$\sqrt{2x-3}=4+\sqrt{x+7}$$

Square both sides of the equation.

$$\left(\sqrt{2x-3}\right)^{2}=\left(4+\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{2x-3} to the power of 2 and get 2x-3.

$$2x-3=\left(4+\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{x+7}\right)^{2}.

$$2x-3=16+8\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

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

$$2x-3=16+8\sqrt{x+7}+x+7$$

Add 16 and 7 to get 23.

$$2x-3=23+8\sqrt{x+7}+x$$

Subtract 23+x from both sides of the equation.

$$2x-3-\left(23+x\right)=8\sqrt{x+7}$$

To find the opposite of 23+x, find the opposite of each term.

$$2x-3-23-x=8\sqrt{x+7}$$

Subtract 23 from -3 to get -26.

$$2x-26-x=8\sqrt{x+7}$$

Combine 2x and -x to get x.

$$x-26=8\sqrt{x+7}$$

Square both sides of the equation.

$$\left(x-26\right)^{2}=\left(8\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-26\right)^{2}.

$$x^{2}-52x+676=\left(8\sqrt{x+7}\right)^{2}$$

Expand \left(8\sqrt{x+7}\right)^{2}.

$$x^{2}-52x+676=8^{2}\left(\sqrt{x+7}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$x^{2}-52x+676=64\left(\sqrt{x+7}\right)^{2}$$

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

$$x^{2}-52x+676=64\left(x+7\right)$$

Use the distributive property to multiply 64 by x+7.

$$x^{2}-52x+676=64x+448$$

Subtract 64x from both sides.

$$x^{2}-52x+676-64x=448$$

Combine -52x and -64x to get -116x.

$$x^{2}-116x+676=448$$

Subtract 448 from both sides.

$$x^{2}-116x+676-448=0$$

Subtract 448 from 676 to get 228.

$$x^{2}-116x+228=0$$

To solve the equation, factor x^{2}-116x+228 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.

$$a+b=-116 ab=228$$

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 228.

$$-1,-228 -2,-114 -3,-76 -4,-57 -6,-38 -12,-19$$

Calculate the sum for each pair.

$$-1-228=-229 -2-114=-116 -3-76=-79 -4-57=-61 -6-38=-44 -12-19=-31$$

The solution is the pair that gives sum -116.

$$a=-114 b=-2$$

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

$$\left(x-114\right)\left(x-2\right)$$

To find equation solutions, solve x-114=0 and x-2=0.

$$x=114 x=2$$

Substitute 114 for x in the equation \sqrt{2x-3}-\sqrt{x+7}=4.

$$\sqrt{2\times 114-3}-\sqrt{114+7}=4$$

Simplify. The value x=114 satisfies the equation.

$$4=4$$

Substitute 2 for x in the equation \sqrt{2x-3}-\sqrt{x+7}=4.

$$\sqrt{2\times 2-3}-\sqrt{2+7}=4$$

Simplify. The value x=2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=4$$

Substitute 114 for x in the equation \sqrt{2x-3}-\sqrt{x+7}=4.

$$\sqrt{2\times 114-3}-\sqrt{114+7}=4$$

Simplify. The value x=114 satisfies the equation.

$$4=4$$

Equation \sqrt{2x-3}=\sqrt{x+7}+4 has a unique solution.

$$x=114$$