Solve x=x^2-6x+14

$$x = x ^ { 2 } - 6 x + 14$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{7}i+7}{2}\approx 3.5-1.322875656i$$

$$x=\frac{7+\sqrt{7}i}{2}\approx 3.5+1.322875656i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-x^{2}=-6x+14$$

Add 6x to both sides.

$$x-x^{2}+6x=14$$

Combine x and 6x to get 7x.

$$7x-x^{2}=14$$

Subtract 14 from both sides.

$$7x-x^{2}-14=0$$

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.

$$-x^{2}+7x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-7±\sqrt{49+4\left(-14\right)}}{2\left(-1\right)}$$

Multiply 4 times -14.

$$x=\frac{-7±\sqrt{49-56}}{2\left(-1\right)}$$

Add 49 to -56.

$$x=\frac{-7±\sqrt{-7}}{2\left(-1\right)}$$

Take the square root of -7.

$$x=\frac{-7±\sqrt{7}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-7±\sqrt{7}i}{-2}$$

Now solve the equation x=\frac{-7±\sqrt{7}i}{-2} when ± is plus. Add -7 to i\sqrt{7}.

$$x=\frac{-7+\sqrt{7}i}{-2}$$

Divide -7+i\sqrt{7} by -2.

$$x=\frac{-\sqrt{7}i+7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{7}i}{-2} when ± is minus. Subtract i\sqrt{7} from -7.

$$x=\frac{-\sqrt{7}i-7}{-2}$$

Divide -7-i\sqrt{7} by -2.

$$x=\frac{7+\sqrt{7}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{7}i+7}{2} x=\frac{7+\sqrt{7}i}{2}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-x^{2}=-6x+14$$

Add 6x to both sides.

$$x-x^{2}+6x=14$$

Combine x and 6x to get 7x.

$$7x-x^{2}=14$$

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.

$$-x^{2}+7x=14$$

Divide both sides by -1.

$$\frac{-x^{2}+7x}{-1}=\frac{14}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{7}{-1}x=\frac{14}{-1}$$

Divide 7 by -1.

$$x^{2}-7x=\frac{14}{-1}$$

Divide 14 by -1.

$$x^{2}-7x=-14$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-14+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-14+\frac{49}{4}$$

Add -14 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=-\frac{7}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{7}i}{2} x-\frac{7}{2}=-\frac{\sqrt{7}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{7+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+7}{2}$$