Solve -18=x-x^2

$$-18 = x- { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{73} + 1}{2} \approx 4.772001873$$

$$x=\frac{1-\sqrt{73}}{2}\approx -3.772001873$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x-x^{2}=-18$$

Add 18 to both sides.

$$x-x^{2}+18=0$$

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.

$$-x^{2}+x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 18}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1±\sqrt{1+4\times 18}}{2\left(-1\right)}$$

Multiply 4 times 18.

$$x=\frac{-1±\sqrt{1+72}}{2\left(-1\right)}$$

Add 1 to 72.

$$x=\frac{-1±\sqrt{73}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\sqrt{73}}{-2}$$

Now solve the equation x=\frac{-1±\sqrt{73}}{-2} when ± is plus. Add -1 to \sqrt{73}.

$$x=\frac{\sqrt{73}-1}{-2}$$

Divide -1+\sqrt{73} by -2.

$$x=\frac{1-\sqrt{73}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{73}}{-2} when ± is minus. Subtract \sqrt{73} from -1.

$$x=\frac{-\sqrt{73}-1}{-2}$$

Divide -1-\sqrt{73} by -2.

$$x=\frac{\sqrt{73}+1}{2}$$

The equation is now solved.

$$x=\frac{1-\sqrt{73}}{2} x=\frac{\sqrt{73}+1}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x-x^{2}=-18$$

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.

$$-x^{2}+x=-18$$

Divide both sides by -1.

$$\frac{-x^{2}+x}{-1}=-\frac{18}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1}{-1}x=-\frac{18}{-1}$$

Divide 1 by -1.

$$x^{2}-x=-\frac{18}{-1}$$

Divide -18 by -1.

$$x^{2}-x=18$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=18+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=18+\frac{1}{4}$$

Add 18 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{73}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{73}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{73}}{2} x-\frac{1}{2}=-\frac{\sqrt{73}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{73}+1}{2} x=\frac{1-\sqrt{73}}{2}$$