Solve x=-13x^2

$$x = - 13 x ^ { 2 }$$

Solve for x

$$x=-\frac{1}{13}\approx -0.076923077$$

$$x=0$$

Steps Using Factoring

Add 13x^{2} to both sides.

$$x+13x^{2}=0$$

Factor out x.

$$x\left(1+13x\right)=0$$

To find equation solutions, solve x=0 and 1+13x=0.

$$x=0 x=-\frac{1}{13}$$

Steps Using the Quadratic Formula

Add 13x^{2} to both sides.

$$x+13x^{2}=0$$

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.

$$13x^{2}+x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}}}{2\times 13}$$

Take the square root of 1^{2}.

$$x=\frac{-1±1}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{-1±1}{26}$$

Now solve the equation x=\frac{-1±1}{26} when ± is plus. Add -1 to 1.

$$x=\frac{0}{26}$$

Divide 0 by 26.

$$x=0$$

Now solve the equation x=\frac{-1±1}{26} when ± is minus. Subtract 1 from -1.

$$x=-\frac{2}{26}$$

Reduce the fraction \frac{-2}{26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{13}$$

The equation is now solved.

$$x=0 x=-\frac{1}{13}$$

Steps for Completing the Square

Add 13x^{2} to both sides.

$$x+13x^{2}=0$$

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.

$$13x^{2}+x=0$$

Divide both sides by 13.

$$\frac{13x^{2}+x}{13}=\frac{0}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}+\frac{1}{13}x=\frac{0}{13}$$

Divide 0 by 13.

$$x^{2}+\frac{1}{13}x=0$$

Divide \frac{1}{13}, the coefficient of the x term, by 2 to get \frac{1}{26}. Then add the square of \frac{1}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{13}x+\left(\frac{1}{26}\right)^{2}=\left(\frac{1}{26}\right)^{2}$$

Square \frac{1}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{13}x+\frac{1}{676}=\frac{1}{676}$$

Factor x^{2}+\frac{1}{13}x+\frac{1}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{26}\right)^{2}=\frac{1}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{26}\right)^{2}}=\sqrt{\frac{1}{676}}$$

Simplify.

$$x+\frac{1}{26}=\frac{1}{26} x+\frac{1}{26}=-\frac{1}{26}$$

Subtract \frac{1}{26} from both sides of the equation.

$$x=0 x=-\frac{1}{13}$$