Solve sqrt{2x}-9=x-4

$$\sqrt { 2 x } - 9 = x - 4$$

Solve for x (complex solution)

$$x=-4+3i$$

$$x=-4-3i$$

Solution Steps

Subtract -9 from both sides of the equation.

$$\sqrt{2x}=x-4+9$$

Add -4 and 9 to get 5.

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

Square both sides of the equation.

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

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

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

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

$$2x=x^{2}+10x+25$$

Subtract x^{2} from both sides.

$$2x-x^{2}=10x+25$$

Subtract 10x from both sides.

$$2x-x^{2}-10x=25$$

Combine 2x and -10x to get -8x.

$$-8x-x^{2}=25$$

Subtract 25 from both sides.

$$-8x-x^{2}-25=0$$

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.

$$-x^{2}-8x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\left(-25\right)}}{2\left(-1\right)}$$

Multiply 4 times -25.

$$x=\frac{-\left(-8\right)±\sqrt{64-100}}{2\left(-1\right)}$$

Add 64 to -100.

$$x=\frac{-\left(-8\right)±\sqrt{-36}}{2\left(-1\right)}$$

Take the square root of -36.

$$x=\frac{-\left(-8\right)±6i}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±6i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±6i}{-2}$$

Now solve the equation x=\frac{8±6i}{-2} when ± is plus. Add 8 to 6i.

$$x=\frac{8+6i}{-2}$$

Divide 8+6i by -2.

$$x=-4-3i$$

Now solve the equation x=\frac{8±6i}{-2} when ± is minus. Subtract 6i from 8.

$$x=\frac{8-6i}{-2}$$

Divide 8-6i by -2.

$$x=-4+3i$$

The equation is now solved.

$$x=-4-3i x=-4+3i$$

Substitute -4-3i for x in the equation \sqrt{2x}-9=x-4.

$$\sqrt{2\left(-4-3i\right)}-9=-4-3i-4$$

Simplify. The value x=-4-3i satisfies the equation.

$$-8-3i=-8-3i$$

Substitute -4+3i for x in the equation \sqrt{2x}-9=x-4.

$$\sqrt{2\left(-4+3i\right)}-9=-4+3i-4$$

Simplify. The value x=-4+3i satisfies the equation.

$$-8+3i=-8+3i$$

List all solutions of \sqrt{2x}=x+5.

$$x=-4-3i x=-4+3i$$