Solve (x+1)=9x^2

$$( x + 1 ) = 9 x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{37}+1}{18}\approx 0.393486807$$

$$x=\frac{1-\sqrt{37}}{18}\approx -0.282375696$$

Steps Using the Quadratic Formula

Subtract 9x^{2} from both sides.

$$x+1-9x^{2}=0$$

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.

$$-9x^{2}+x+1=0$$

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

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

Square 1.

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

Multiply -4 times -9.

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

Add 1 to 36.

$$x=\frac{-1±\sqrt{37}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-1±\sqrt{37}}{-18}$$

Now solve the equation x=\frac{-1±\sqrt{37}}{-18} when ± is plus. Add -1 to \sqrt{37}.

$$x=\frac{\sqrt{37}-1}{-18}$$

Divide -1+\sqrt{37} by -18.

$$x=\frac{1-\sqrt{37}}{18}$$

Now solve the equation x=\frac{-1±\sqrt{37}}{-18} when ± is minus. Subtract \sqrt{37} from -1.

$$x=\frac{-\sqrt{37}-1}{-18}$$

Divide -1-\sqrt{37} by -18.

$$x=\frac{\sqrt{37}+1}{18}$$

The equation is now solved.

$$x=\frac{1-\sqrt{37}}{18} x=\frac{\sqrt{37}+1}{18}$$

Steps for Completing the Square

Subtract 9x^{2} from both sides.

$$x+1-9x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x-9x^{2}=-1$$

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.

$$-9x^{2}+x=-1$$

Divide both sides by -9.

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

Dividing by -9 undoes the multiplication by -9.

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

Divide 1 by -9.

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

Divide -1 by -9.

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

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.

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

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

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

Add \frac{1}{9} to \frac{1}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{37}{324}$$

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}.

$$\left(x-\frac{1}{18}\right)^{2}=\frac{37}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{18}\right)^{2}}=\sqrt{\frac{37}{324}}$$

Simplify.

$$x-\frac{1}{18}=\frac{\sqrt{37}}{18} x-\frac{1}{18}=-\frac{\sqrt{37}}{18}$$

Add \frac{1}{18} to both sides of the equation.

$$x=\frac{\sqrt{37}+1}{18} x=\frac{1-\sqrt{37}}{18}$$