Solve x^2+6x/x^2+x-1=0

$$\frac { x ^ { 2 } + 6 x } { x ^ { 2 } + x - 1 } = 0$$

Solve for x

$$x=-6$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by \left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right).

$$x^{2}+6x=0$$

Factor out x.

$$x\left(x+6\right)=0$$

To find equation solutions, solve x=0 and x+6=0.

$$x=0 x=-6$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by \left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right).

$$x^{2}+6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}}}{2}$$

Take the square root of 6^{2}.

$$x=\frac{-6±6}{2}$$

Now solve the equation x=\frac{-6±6}{2} when ± is plus. Add -6 to 6.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

Now solve the equation x=\frac{-6±6}{2} when ± is minus. Subtract 6 from -6.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

The equation is now solved.

$$x=0 x=-6$$

Steps for Completing the Square

Multiply both sides of the equation by \left(x-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right).

$$x^{2}+6x=0$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=3^{2}$$

Square 3.

$$x^{2}+6x+9=9$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+3=3 x+3=-3$$

Subtract 3 from both sides of the equation.

$$x=0 x=-6$$