Solve x=-8(1+sqrt{-2x)}

$$x=-8(1+ \sqrt{ -2x) }$$

Solve for x

$$x=-32\sqrt{5}-72\approx -143.55417528$$

Solution Steps

Use the distributive property to multiply -8 by 1+\sqrt{-2x}.

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

Add 8\sqrt{-2x} to both sides.

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

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Calculate 8 to the power of 2 and get 64.

$$64\left(\sqrt{-2x}\right)^{2}=\left(-8-x\right)^{2}$$

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

$$64\left(-2\right)x=\left(-8-x\right)^{2}$$

Multiply 64 and -2 to get -128.

$$-128x=\left(-8-x\right)^{2}$$

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

$$-128x=64+16x+x^{2}$$

Subtract 16x from both sides.

$$-128x-16x=64+x^{2}$$

Combine -128x and -16x to get -144x.

$$-144x=64+x^{2}$$

Subtract x^{2} from both sides.

$$-144x-x^{2}=64$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-144x=64$$

Subtract 64 from both sides of the equation.

$$-x^{2}-144x-64=64-64$$

Subtracting 64 from itself leaves 0.

$$-x^{2}-144x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -144 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\left(-1\right)\left(-64\right)}}{2\left(-1\right)}$$

Square -144.

$$x=\frac{-\left(-144\right)±\sqrt{20736-4\left(-1\right)\left(-64\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-144\right)±\sqrt{20736+4\left(-64\right)}}{2\left(-1\right)}$$

Multiply 4 times -64.

$$x=\frac{-\left(-144\right)±\sqrt{20736-256}}{2\left(-1\right)}$$

Add 20736 to -256.

$$x=\frac{-\left(-144\right)±\sqrt{20480}}{2\left(-1\right)}$$

Take the square root of 20480.

$$x=\frac{-\left(-144\right)±64\sqrt{5}}{2\left(-1\right)}$$

The opposite of -144 is 144.

$$x=\frac{144±64\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{144±64\sqrt{5}}{-2}$$

Now solve the equation x=\frac{144±64\sqrt{5}}{-2} when ± is plus. Add 144 to 64\sqrt{5}.

$$x=\frac{64\sqrt{5}+144}{-2}$$

Divide 144+64\sqrt{5} by -2.

$$x=-32\sqrt{5}-72$$

Now solve the equation x=\frac{144±64\sqrt{5}}{-2} when ± is minus. Subtract 64\sqrt{5} from 144.

$$x=\frac{144-64\sqrt{5}}{-2}$$

Divide 144-64\sqrt{5} by -2.

$$x=32\sqrt{5}-72$$

The equation is now solved.

$$x=-32\sqrt{5}-72 x=32\sqrt{5}-72$$

Substitute -32\sqrt{5}-72 for x in the equation x=-8\left(1+\sqrt{-2x}\right).

$$-32\sqrt{5}-72=-8\left(1+\sqrt{-2\left(-32\sqrt{5}-72\right)}\right)$$

Simplify. The value x=-32\sqrt{5}-72 satisfies the equation.

$$-32\times 5^{\frac{1}{2}}-72=-72-32\times 5^{\frac{1}{2}}$$

Substitute 32\sqrt{5}-72 for x in the equation x=-8\left(1+\sqrt{-2x}\right).

$$32\sqrt{5}-72=-8\left(1+\sqrt{-2\left(32\sqrt{5}-72\right)}\right)$$

Simplify. The value x=32\sqrt{5}-72 does not satisfy the equation.

$$32\times 5^{\frac{1}{2}}-72=56-32\times 5^{\frac{1}{2}}$$

Equation 8\sqrt{-2x}=-x-8 has a unique solution.

$$x=-32\sqrt{5}-72$$