Solve 30-x=9x^2

$$30-x=9 { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{1081} - 1}{18} \approx 1.771031358$$

$$x=\frac{-\sqrt{1081}-1}{18}\approx -1.882142469$$

Steps Using the Quadratic Formula

Subtract 9x^{2} from both sides.

$$30-x-9x^{2}=0$$

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.

$$-9x^{2}-x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -1 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-9\right)\times 30}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-1\right)±\sqrt{1+36\times 30}}{2\left(-9\right)}$$

Multiply 36 times 30.

$$x=\frac{-\left(-1\right)±\sqrt{1+1080}}{2\left(-9\right)}$$

Add 1 to 1080.

$$x=\frac{-\left(-1\right)±\sqrt{1081}}{2\left(-9\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{1081}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{1±\sqrt{1081}}{-18}$$

Now solve the equation x=\frac{1±\sqrt{1081}}{-18} when ± is plus. Add 1 to \sqrt{1081}.

$$x=\frac{\sqrt{1081}+1}{-18}$$

Divide 1+\sqrt{1081} by -18.

$$x=\frac{-\sqrt{1081}-1}{18}$$

Now solve the equation x=\frac{1±\sqrt{1081}}{-18} when ± is minus. Subtract \sqrt{1081} from 1.

$$x=\frac{1-\sqrt{1081}}{-18}$$

Divide 1-\sqrt{1081} by -18.

$$x=\frac{\sqrt{1081}-1}{18}$$

The equation is now solved.

$$x=\frac{-\sqrt{1081}-1}{18} x=\frac{\sqrt{1081}-1}{18}$$

Steps for Completing the Square

Subtract 9x^{2} from both sides.

$$30-x-9x^{2}=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$-x-9x^{2}=-30$$

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.

$$-9x^{2}-x=-30$$

Divide both sides by -9.

$$\frac{-9x^{2}-x}{-9}=-\frac{30}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{1}{-9}\right)x=-\frac{30}{-9}$$

Divide -1 by -9.

$$x^{2}+\frac{1}{9}x=-\frac{30}{-9}$$

Reduce the fraction \frac{-30}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{9}x=\frac{10}{3}$$

Divide \frac{1}{9}, the coefficient of the x term, by 2 to get \frac{1}{18}. Then add the square of \frac{1}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{9}x+\left(\frac{1}{18}\right)^{2}=\frac{10}{3}+\left(\frac{1}{18}\right)^{2}$$

Square \frac{1}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{9}x+\frac{1}{324}=\frac{10}{3}+\frac{1}{324}$$

Add \frac{10}{3} to \frac{1}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{9}x+\frac{1}{324}=\frac{1081}{324}$$

Factor x^{2}+\frac{1}{9}x+\frac{1}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{18}\right)^{2}=\frac{1081}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{18}\right)^{2}}=\sqrt{\frac{1081}{324}}$$

Simplify.

$$x+\frac{1}{18}=\frac{\sqrt{1081}}{18} x+\frac{1}{18}=-\frac{\sqrt{1081}}{18}$$

Subtract \frac{1}{18} from both sides of the equation.

$$x=\frac{\sqrt{1081}-1}{18} x=\frac{-\sqrt{1081}-1}{18}$$