Solve -7x=2x^2

$$- 7 x = 2 x ^ { 2 }$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=0$$

Steps Using Factoring

Subtract 2x^{2} from both sides.

$$-7x-2x^{2}=0$$

Factor out x.

$$x\left(-7-2x\right)=0$$

To find equation solutions, solve x=0 and -7-2x=0.

$$x=0 x=-\frac{7}{2}$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$-7x-2x^{2}=0$$

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.

$$-2x^{2}-7x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}}}{2\left(-2\right)}$$

Take the square root of \left(-7\right)^{2}.

$$x=\frac{-\left(-7\right)±7}{2\left(-2\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{7±7}{-4}$$

Now solve the equation x=\frac{7±7}{-4} when ± is plus. Add 7 to 7.

$$x=\frac{14}{-4}$$

Reduce the fraction \frac{14}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{2}$$

Now solve the equation x=\frac{7±7}{-4} when ± is minus. Subtract 7 from 7.

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

Divide 0 by -4.

$$x=0$$

The equation is now solved.

$$x=-\frac{7}{2} x=0$$

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$-7x-2x^{2}=0$$

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.

$$-2x^{2}-7x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}-7x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{7}{-2}\right)x=\frac{0}{-2}$$

Divide -7 by -2.

$$x^{2}+\frac{7}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}+\frac{7}{2}x=0$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{7}{4} x+\frac{7}{4}=-\frac{7}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=0 x=-\frac{7}{2}$$