Solve x/4=1x/2left(1/3x-2right)

$$\frac{ x }{ 4 } = \frac{ 1x }{ 2 } \left( \frac{ 1 }{ 3 } x-2 \right)$$

Solve for x

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 12, the least common multiple of 4,2,3.

$$3x=6\times 1x\left(\frac{1}{3}x-2\right)$$

Multiply 6 and 1 to get 6.

$$3x=6x\left(\frac{1}{3}x-2\right)$$

Use the distributive property to multiply 6x by \frac{1}{3}x-2.

$$3x=6x\times \frac{1}{3}x-12x$$

Multiply x and x to get x^{2}.

$$3x=6x^{2}\times \frac{1}{3}-12x$$

Multiply 6 and \frac{1}{3} to get \frac{6}{3}.

$$3x=\frac{6}{3}x^{2}-12x$$

Divide 6 by 3 to get 2.

$$3x=2x^{2}-12x$$

Subtract 2x^{2} from both sides.

$$3x-2x^{2}=-12x$$

Add 12x to both sides.

$$3x-2x^{2}+12x=0$$

Combine 3x and 12x to get 15x.

$$15x-2x^{2}=0$$

Factor out x.

$$x\left(15-2x\right)=0$$

To find equation solutions, solve x=0 and 15-2x=0.

$$x=0 x=\frac{15}{2}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 4,2,3.

$$3x=6\times 1x\left(\frac{1}{3}x-2\right)$$

Multiply 6 and 1 to get 6.

$$3x=6x\left(\frac{1}{3}x-2\right)$$

Use the distributive property to multiply 6x by \frac{1}{3}x-2.

$$3x=6x\times \frac{1}{3}x-12x$$

Multiply x and x to get x^{2}.

$$3x=6x^{2}\times \frac{1}{3}-12x$$

Multiply 6 and \frac{1}{3} to get \frac{6}{3}.

$$3x=\frac{6}{3}x^{2}-12x$$

Divide 6 by 3 to get 2.

$$3x=2x^{2}-12x$$

Subtract 2x^{2} from both sides.

$$3x-2x^{2}=-12x$$

Add 12x to both sides.

$$3x-2x^{2}+12x=0$$

Combine 3x and 12x to get 15x.

$$15x-2x^{2}=0$$

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.

$$-2x^{2}+15x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 15 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}}}{2\left(-2\right)}$$

Take the square root of 15^{2}.

$$x=\frac{-15±15}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-15±15}{-4}$$

Now solve the equation x=\frac{-15±15}{-4} when ± is plus. Add -15 to 15.

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

Divide 0 by -4.

$$x=0$$

Now solve the equation x=\frac{-15±15}{-4} when ± is minus. Subtract 15 from -15.

$$x=-\frac{30}{-4}$$

Reduce the fraction \frac{-30}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{15}{2}$$

The equation is now solved.

$$x=0 x=\frac{15}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 4,2,3.

$$3x=6\times 1x\left(\frac{1}{3}x-2\right)$$

Multiply 6 and 1 to get 6.

$$3x=6x\left(\frac{1}{3}x-2\right)$$

Use the distributive property to multiply 6x by \frac{1}{3}x-2.

$$3x=6x\times \frac{1}{3}x-12x$$

Multiply x and x to get x^{2}.

$$3x=6x^{2}\times \frac{1}{3}-12x$$

Multiply 6 and \frac{1}{3} to get \frac{6}{3}.

$$3x=\frac{6}{3}x^{2}-12x$$

Divide 6 by 3 to get 2.

$$3x=2x^{2}-12x$$

Subtract 2x^{2} from both sides.

$$3x-2x^{2}=-12x$$

Add 12x to both sides.

$$3x-2x^{2}+12x=0$$

Combine 3x and 12x to get 15x.

$$15x-2x^{2}=0$$

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.

$$-2x^{2}+15x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+15x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{15}{-2}x=\frac{0}{-2}$$

Divide 15 by -2.

$$x^{2}-\frac{15}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}-\frac{15}{2}x=0$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=\frac{225}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{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}.

$$\left(x-\frac{15}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{15}{4} x-\frac{15}{4}=-\frac{15}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$x=\frac{15}{2} x=0$$