Solve left(120-x-1/3left(120-xright)right)x-10x=0

$$\left( 120-x- \frac{ 1 }{ 3 } \left( 120-x \right) \right) x-10x=0$$

Solve for x

$$x=105$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply -\frac{1}{3} by 120-x.

$$\left(120-x-\frac{1}{3}\times 120-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Express -\frac{1}{3}\times 120 as a single fraction.

$$\left(120-x+\frac{-120}{3}-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Divide -120 by 3 to get -40.

$$\left(120-x-40-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

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

$$\left(120-x-40+\frac{1}{3}x\right)x-10x=0$$

Subtract 40 from 120 to get 80.

$$\left(80-x+\frac{1}{3}x\right)x-10x=0$$

Combine -x and \frac{1}{3}x to get -\frac{2}{3}x.

$$\left(80-\frac{2}{3}x\right)x-10x=0$$

Use the distributive property to multiply 80-\frac{2}{3}x by x.

$$80x-\frac{2}{3}xx-10x=0$$

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

$$80x-\frac{2}{3}x^{2}-10x=0$$

Combine 80x and -10x to get 70x.

$$70x-\frac{2}{3}x^{2}=0$$

Factor out x.

$$x\left(70-\frac{2}{3}x\right)=0$$

To find equation solutions, solve x=0 and 70-\frac{2x}{3}=0.

$$x=0 x=105$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -\frac{1}{3} by 120-x.

$$\left(120-x-\frac{1}{3}\times 120-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Express -\frac{1}{3}\times 120 as a single fraction.

$$\left(120-x+\frac{-120}{3}-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Divide -120 by 3 to get -40.

$$\left(120-x-40-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

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

$$\left(120-x-40+\frac{1}{3}x\right)x-10x=0$$

Subtract 40 from 120 to get 80.

$$\left(80-x+\frac{1}{3}x\right)x-10x=0$$

Combine -x and \frac{1}{3}x to get -\frac{2}{3}x.

$$\left(80-\frac{2}{3}x\right)x-10x=0$$

Use the distributive property to multiply 80-\frac{2}{3}x by x.

$$80x-\frac{2}{3}xx-10x=0$$

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

$$80x-\frac{2}{3}x^{2}-10x=0$$

Combine 80x and -10x to get 70x.

$$70x-\frac{2}{3}x^{2}=0$$

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.

$$-\frac{2}{3}x^{2}+70x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{3} for a, 70 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}}}{2\left(-\frac{2}{3}\right)}$$

Take the square root of 70^{2}.

$$x=\frac{-70±70}{2\left(-\frac{2}{3}\right)}$$

Multiply 2 times -\frac{2}{3}.

$$x=\frac{-70±70}{-\frac{4}{3}}$$

Now solve the equation x=\frac{-70±70}{-\frac{4}{3}} when ± is plus. Add -70 to 70.

$$x=\frac{0}{-\frac{4}{3}}$$

Divide 0 by -\frac{4}{3} by multiplying 0 by the reciprocal of -\frac{4}{3}.

$$x=0$$

Now solve the equation x=\frac{-70±70}{-\frac{4}{3}} when ± is minus. Subtract 70 from -70.

$$x=-\frac{140}{-\frac{4}{3}}$$

Divide -140 by -\frac{4}{3} by multiplying -140 by the reciprocal of -\frac{4}{3}.

$$x=105$$

The equation is now solved.

$$x=0 x=105$$

Steps for Completing the Square

Use the distributive property to multiply -\frac{1}{3} by 120-x.

$$\left(120-x-\frac{1}{3}\times 120-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Express -\frac{1}{3}\times 120 as a single fraction.

$$\left(120-x+\frac{-120}{3}-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

Divide -120 by 3 to get -40.

$$\left(120-x-40-\frac{1}{3}\left(-1\right)x\right)x-10x=0$$

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

$$\left(120-x-40+\frac{1}{3}x\right)x-10x=0$$

Subtract 40 from 120 to get 80.

$$\left(80-x+\frac{1}{3}x\right)x-10x=0$$

Combine -x and \frac{1}{3}x to get -\frac{2}{3}x.

$$\left(80-\frac{2}{3}x\right)x-10x=0$$

Use the distributive property to multiply 80-\frac{2}{3}x by x.

$$80x-\frac{2}{3}xx-10x=0$$

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

$$80x-\frac{2}{3}x^{2}-10x=0$$

Combine 80x and -10x to get 70x.

$$70x-\frac{2}{3}x^{2}=0$$

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.

$$-\frac{2}{3}x^{2}+70x=0$$

Divide both sides of the equation by -\frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{3}x^{2}+70x}{-\frac{2}{3}}=\frac{0}{-\frac{2}{3}}$$

Dividing by -\frac{2}{3} undoes the multiplication by -\frac{2}{3}.

$$x^{2}+\frac{70}{-\frac{2}{3}}x=\frac{0}{-\frac{2}{3}}$$

Divide 70 by -\frac{2}{3} by multiplying 70 by the reciprocal of -\frac{2}{3}.

$$x^{2}-105x=\frac{0}{-\frac{2}{3}}$$

Divide 0 by -\frac{2}{3} by multiplying 0 by the reciprocal of -\frac{2}{3}.

$$x^{2}-105x=0$$

Divide -105, the coefficient of the x term, by 2 to get -\frac{105}{2}. Then add the square of -\frac{105}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-105x+\left(-\frac{105}{2}\right)^{2}=\left(-\frac{105}{2}\right)^{2}$$

Square -\frac{105}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-105x+\frac{11025}{4}=\frac{11025}{4}$$

Factor x^{2}-105x+\frac{11025}{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}.

$$\left(x-\frac{105}{2}\right)^{2}=\frac{11025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{105}{2}\right)^{2}}=\sqrt{\frac{11025}{4}}$$

Simplify.

$$x-\frac{105}{2}=\frac{105}{2} x-\frac{105}{2}=-\frac{105}{2}$$

Add \frac{105}{2} to both sides of the equation.

$$x=105 x=0$$