Solve sqrt{2x-15}-sqrt{x+16}=-1

$$\sqrt { 2 x - 15 } - \sqrt { x + 16 } = - 1$$

Solve for x

$$x=20$$

Solution Steps

Subtract -\sqrt{x+16} from both sides of the equation.

$$\sqrt{2x-15}=-1+\sqrt{x+16}$$

Square both sides of the equation.

$$\left(\sqrt{2x-15}\right)^{2}=\left(-1+\sqrt{x+16}\right)^{2}$$

Calculate \sqrt{2x-15} to the power of 2 and get 2x-15.

$$2x-15=\left(-1+\sqrt{x+16}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1+\sqrt{x+16}\right)^{2}.

$$2x-15=1-2\sqrt{x+16}+\left(\sqrt{x+16}\right)^{2}$$

Calculate \sqrt{x+16} to the power of 2 and get x+16.

$$2x-15=1-2\sqrt{x+16}+x+16$$

Add 1 and 16 to get 17.

$$2x-15=17-2\sqrt{x+16}+x$$

Subtract 17+x from both sides of the equation.

$$2x-15-\left(17+x\right)=-2\sqrt{x+16}$$

To find the opposite of 17+x, find the opposite of each term.

$$2x-15-17-x=-2\sqrt{x+16}$$

Subtract 17 from -15 to get -32.

$$2x-32-x=-2\sqrt{x+16}$$

Combine 2x and -x to get x.

$$x-32=-2\sqrt{x+16}$$

Square both sides of the equation.

$$\left(x-32\right)^{2}=\left(-2\sqrt{x+16}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-32\right)^{2}.

$$x^{2}-64x+1024=\left(-2\sqrt{x+16}\right)^{2}$$

Expand \left(-2\sqrt{x+16}\right)^{2}.

$$x^{2}-64x+1024=\left(-2\right)^{2}\left(\sqrt{x+16}\right)^{2}$$

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

$$x^{2}-64x+1024=4\left(\sqrt{x+16}\right)^{2}$$

Calculate \sqrt{x+16} to the power of 2 and get x+16.

$$x^{2}-64x+1024=4\left(x+16\right)$$

Use the distributive property to multiply 4 by x+16.

$$x^{2}-64x+1024=4x+64$$

Subtract 4x from both sides.

$$x^{2}-64x+1024-4x=64$$

Combine -64x and -4x to get -68x.

$$x^{2}-68x+1024=64$$

Subtract 64 from both sides.

$$x^{2}-68x+1024-64=0$$

Subtract 64 from 1024 to get 960.

$$x^{2}-68x+960=0$$

To solve the equation, factor x^{2}-68x+960 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.

$$a+b=-68 ab=960$$

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 960.

$$-1,-960 -2,-480 -3,-320 -4,-240 -5,-192 -6,-160 -8,-120 -10,-96 -12,-80 -15,-64 -16,-60 -20,-48 -24,-40 -30,-32$$

Calculate the sum for each pair.

$$-1-960=-961 -2-480=-482 -3-320=-323 -4-240=-244 -5-192=-197 -6-160=-166 -8-120=-128 -10-96=-106 -12-80=-92 -15-64=-79 -16-60=-76 -20-48=-68 -24-40=-64 -30-32=-62$$

The solution is the pair that gives sum -68.

$$a=-48 b=-20$$

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

$$\left(x-48\right)\left(x-20\right)$$

To find equation solutions, solve x-48=0 and x-20=0.

$$x=48 x=20$$

Substitute 48 for x in the equation \sqrt{2x-15}-\sqrt{x+16}=-1.

$$\sqrt{2\times 48-15}-\sqrt{48+16}=-1$$

Simplify. The value x=48 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Substitute 20 for x in the equation \sqrt{2x-15}-\sqrt{x+16}=-1.

$$\sqrt{2\times 20-15}-\sqrt{20+16}=-1$$

Simplify. The value x=20 satisfies the equation.

$$-1=-1$$

Equation \sqrt{2x-15}=\sqrt{x+16}-1 has a unique solution.

$$x=20$$