Solve x(x-6sqrt{2})+65=0

$$x(x-6 \sqrt{ 2 } )+65=0$$

Solve for x (complex solution)

$$x=3\sqrt{2}+\sqrt{47}i\approx 4.242640687+6.8556546i$$

$$x=-\sqrt{47}i+3\sqrt{2}\approx 4.242640687-6.8556546i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x-6\sqrt{2}.

$$x^{2}-6x\sqrt{2}+65=0$$

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.

$$x^{2}+\left(-6\sqrt{2}\right)x+65=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6\sqrt{2} for b, and 65 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{\left(-6\sqrt{2}\right)^{2}-4\times 65}}{2}$$

Square -6\sqrt{2}.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-4\times 65}}{2}$$

Multiply -4 times 65.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{72-260}}{2}$$

Add 72 to -260.

$$x=\frac{-\left(-6\sqrt{2}\right)±\sqrt{-188}}{2}$$

Take the square root of -188.

$$x=\frac{-\left(-6\sqrt{2}\right)±2\sqrt{47}i}{2}$$

The opposite of -6\sqrt{2} is 6\sqrt{2}.

$$x=\frac{6\sqrt{2}±2\sqrt{47}i}{2}$$

Now solve the equation x=\frac{6\sqrt{2}±2\sqrt{47}i}{2} when ± is plus. Add 6\sqrt{2} to 2i\sqrt{47}.

$$x=\frac{6\sqrt{2}+2\sqrt{47}i}{2}$$

Divide 6\sqrt{2}+2i\sqrt{47} by 2.

$$x=3\sqrt{2}+\sqrt{47}i$$

Now solve the equation x=\frac{6\sqrt{2}±2\sqrt{47}i}{2} when ± is minus. Subtract 2i\sqrt{47} from 6\sqrt{2}.

$$x=\frac{-2\sqrt{47}i+6\sqrt{2}}{2}$$

Divide 6\sqrt{2}-2i\sqrt{47} by 2.

$$x=-\sqrt{47}i+3\sqrt{2}$$

The equation is now solved.

$$x=3\sqrt{2}+\sqrt{47}i x=-\sqrt{47}i+3\sqrt{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by x-6\sqrt{2}.

$$x^{2}-6x\sqrt{2}+65=0$$

Subtract 65 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-6x\sqrt{2}=-65$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+\left(-6\sqrt{2}\right)x=-65$$

Divide -6\sqrt{2}, the coefficient of the x term, by 2 to get -3\sqrt{2}. Then add the square of -3\sqrt{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\left(-6\sqrt{2}\right)x+\left(-3\sqrt{2}\right)^{2}=-65+\left(-3\sqrt{2}\right)^{2}$$

Square -3\sqrt{2}.

$$x^{2}+\left(-6\sqrt{2}\right)x+18=-65+18$$

Add -65 to 18.

$$x^{2}+\left(-6\sqrt{2}\right)x+18=-47$$

Factor x^{2}+\left(-6\sqrt{2}\right)x+18. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\sqrt{2}\right)^{2}=-47$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\sqrt{2}\right)^{2}}=\sqrt{-47}$$

Simplify.

$$x-3\sqrt{2}=\sqrt{47}i x-3\sqrt{2}=-\sqrt{47}i$$

Add 3\sqrt{2} to both sides of the equation.

$$x=3\sqrt{2}+\sqrt{47}i x=-\sqrt{47}i+3\sqrt{2}$$