Solve x-2sqrt{x}+15=0

$$x - 2 \sqrt { x } + 15 = 0$$

Solve for x (complex solution)

$$x=-13+2\sqrt{14}i\approx -13+7.483314774i$$

Solution Steps

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$x-2\sqrt{x}=-15$$

Subtract x from both sides of the equation.

$$-2\sqrt{x}=-15-x$$

Square both sides of the equation.

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

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

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

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

$$4\left(\sqrt{x}\right)^{2}=\left(-15-x\right)^{2}$$

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

$$4x=\left(-15-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-15-x\right)^{2}.

$$4x=225+30x+x^{2}$$

Subtract 30x from both sides.

$$4x-30x=225+x^{2}$$

Combine 4x and -30x to get -26x.

$$-26x=225+x^{2}$$

Subtract x^{2} from both sides.

$$-26x-x^{2}=225$$

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.

$$-x^{2}-26x=225$$

Subtract 225 from both sides of the equation.

$$-x^{2}-26x-225=225-225$$

Subtracting 225 from itself leaves 0.

$$-x^{2}-26x-225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -26 for b, and -225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-1\right)\left(-225\right)}}{2\left(-1\right)}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-1\right)\left(-225\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-26\right)±\sqrt{676+4\left(-225\right)}}{2\left(-1\right)}$$

Multiply 4 times -225.

$$x=\frac{-\left(-26\right)±\sqrt{676-900}}{2\left(-1\right)}$$

Add 676 to -900.

$$x=\frac{-\left(-26\right)±\sqrt{-224}}{2\left(-1\right)}$$

Take the square root of -224.

$$x=\frac{-\left(-26\right)±4\sqrt{14}i}{2\left(-1\right)}$$

The opposite of -26 is 26.

$$x=\frac{26±4\sqrt{14}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{26±4\sqrt{14}i}{-2}$$

Now solve the equation x=\frac{26±4\sqrt{14}i}{-2} when ± is plus. Add 26 to 4i\sqrt{14}.

$$x=\frac{26+4\sqrt{14}i}{-2}$$

Divide 26+4i\sqrt{14} by -2.

$$x=-2\sqrt{14}i-13$$

Now solve the equation x=\frac{26±4\sqrt{14}i}{-2} when ± is minus. Subtract 4i\sqrt{14} from 26.

$$x=\frac{-4\sqrt{14}i+26}{-2}$$

Divide 26-4i\sqrt{14} by -2.

$$x=-13+2\sqrt{14}i$$

The equation is now solved.

$$x=-2\sqrt{14}i-13 x=-13+2\sqrt{14}i$$

Substitute -2\sqrt{14}i-13 for x in the equation x-2\sqrt{x}+15=0.

$$-2\sqrt{14}i-13-2\sqrt{-2\sqrt{14}i-13}+15=0$$

Simplify. The value x=-2\sqrt{14}i-13 does not satisfy the equation.

$$-4i\times 14^{\frac{1}{2}}+4=0$$

Substitute -13+2\sqrt{14}i for x in the equation x-2\sqrt{x}+15=0.

$$-13+2\sqrt{14}i-2\sqrt{-13+2\sqrt{14}i}+15=0$$

Simplify. The value x=-13+2\sqrt{14}i satisfies the equation.

$$0=0$$

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

$$x=-13+2\sqrt{14}i$$