Solve (x)=3x^2-6x

$$( x ) = 3 x ^ { 2 } - 6 x$$

Solve for x

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

$$x=0$$

Steps Using Factoring

Subtract 3x^{2} from both sides.

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

Add 6x to both sides.

$$x-3x^{2}+6x=0$$

Combine x and 6x to get 7x.

$$7x-3x^{2}=0$$

Factor out x.

$$x\left(7-3x\right)=0$$

To find equation solutions, solve x=0 and 7-3x=0.

$$x=0 x=\frac{7}{3}$$

Steps Using the Quadratic Formula

Subtract 3x^{2} from both sides.

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

Add 6x to both sides.

$$x-3x^{2}+6x=0$$

Combine x and 6x to get 7x.

$$7x-3x^{2}=0$$

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.

$$-3x^{2}+7x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}}}{2\left(-3\right)}$$

Take the square root of 7^{2}.

$$x=\frac{-7±7}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±7}{-6}$$

Now solve the equation x=\frac{-7±7}{-6} when ± is plus. Add -7 to 7.

$$x=\frac{0}{-6}$$

Divide 0 by -6.

$$x=0$$

Now solve the equation x=\frac{-7±7}{-6} when ± is minus. Subtract 7 from -7.

$$x=-\frac{14}{-6}$$

Reduce the fraction \frac{-14}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{3}$$

The equation is now solved.

$$x=0 x=\frac{7}{3}$$

Steps for Completing the Square

Subtract 3x^{2} from both sides.

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

Add 6x to both sides.

$$x-3x^{2}+6x=0$$

Combine x and 6x to get 7x.

$$7x-3x^{2}=0$$

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.

$$-3x^{2}+7x=0$$

Divide both sides by -3.

$$\frac{-3x^{2}+7x}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{7}{-3}x=\frac{0}{-3}$$

Divide 7 by -3.

$$x^{2}-\frac{7}{3}x=\frac{0}{-3}$$

Divide 0 by -3.

$$x^{2}-\frac{7}{3}x=0$$

Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=\left(-\frac{7}{6}\right)^{2}$$

Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{49}{36}$$

Factor x^{2}-\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x-\frac{7}{6}=\frac{7}{6} x-\frac{7}{6}=-\frac{7}{6}$$

Add \frac{7}{6} to both sides of the equation.

$$x=\frac{7}{3} x=0$$