Solve x^2-4x-5x^2=0

$$x ^ { 2 } - 4 x - 5 x ^ { 2 } = 0$$

Solve for x

$$x=-1$$

$$x=0$$

Steps Using the Quadratic Formula

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-4x=0$$

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

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

Take the square root of \left(-4\right)^{2}.

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

The opposite of -4 is 4.

$$x=\frac{4±4}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{4±4}{-8}$$

Now solve the equation x=\frac{4±4}{-8} when ± is plus. Add 4 to 4.

$$x=\frac{8}{-8}$$

Divide 8 by -8.

$$x=-1$$

Now solve the equation x=\frac{4±4}{-8} when ± is minus. Subtract 4 from 4.

$$x=\frac{0}{-8}$$

Divide 0 by -8.

$$x=0$$

The equation is now solved.

$$x=-1 x=0$$

Steps for Completing the Square

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-4x=0$$

Divide both sides by -4.

$$\frac{-4x^{2}-4x}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{4}{-4}\right)x=\frac{0}{-4}$$

Divide -4 by -4.

$$x^{2}+x=\frac{0}{-4}$$

Divide 0 by -4.

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

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.

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=0 x=-1$$