Solve x^2-10x/3x-30=(x)

$$\frac { x ^ { 2 } - 10 x } { 3 x - 30 } = ( x )$$

Solve for x

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-10\right).

$$x^{2}-10x=x\times 3\left(x-10\right)$$

Use the distributive property to multiply x\times 3 by x-10.

$$x^{2}-10x=3x^{2}-10x\times 3$$

Multiply -10 and 3 to get -30.

$$x^{2}-10x=3x^{2}-30x$$

Subtract 3x^{2} from both sides.

$$x^{2}-10x-3x^{2}=-30x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-10x=-30x$$

Add 30x to both sides.

$$-2x^{2}-10x+30x=0$$

Combine -10x and 30x to get 20x.

$$-2x^{2}+20x=0$$

Factor out x.

$$x\left(-2x+20\right)=0$$

To find equation solutions, solve x=0 and -2x+20=0.

$$x=0 x=10$$

Variable x cannot be equal to 10.

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-10\right).

$$x^{2}-10x=x\times 3\left(x-10\right)$$

Use the distributive property to multiply x\times 3 by x-10.

$$x^{2}-10x=3x^{2}-10x\times 3$$

Multiply -10 and 3 to get -30.

$$x^{2}-10x=3x^{2}-30x$$

Subtract 3x^{2} from both sides.

$$x^{2}-10x-3x^{2}=-30x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-10x=-30x$$

Add 30x to both sides.

$$-2x^{2}-10x+30x=0$$

Combine -10x and 30x to get 20x.

$$-2x^{2}+20x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 20 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}}}{2\left(-2\right)}$$

Take the square root of 20^{2}.

$$x=\frac{-20±20}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-20±20}{-4}$$

Now solve the equation x=\frac{-20±20}{-4} when ± is plus. Add -20 to 20.

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

Divide 0 by -4.

$$x=0$$

Now solve the equation x=\frac{-20±20}{-4} when ± is minus. Subtract 20 from -20.

$$x=-\frac{40}{-4}$$

Divide -40 by -4.

$$x=10$$

The equation is now solved.

$$x=0 x=10$$

Variable x cannot be equal to 10.

$$x=0$$

Steps for Completing the Square

Variable x cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-10\right).

$$x^{2}-10x=x\times 3\left(x-10\right)$$

Use the distributive property to multiply x\times 3 by x-10.

$$x^{2}-10x=3x^{2}-10x\times 3$$

Multiply -10 and 3 to get -30.

$$x^{2}-10x=3x^{2}-30x$$

Subtract 3x^{2} from both sides.

$$x^{2}-10x-3x^{2}=-30x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-10x=-30x$$

Add 30x to both sides.

$$-2x^{2}-10x+30x=0$$

Combine -10x and 30x to get 20x.

$$-2x^{2}+20x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+20x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{20}{-2}x=\frac{0}{-2}$$

Divide 20 by -2.

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

Divide 0 by -2.

$$x^{2}-10x=0$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=25$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-5=5 x-5=-5$$

Add 5 to both sides of the equation.

$$x=10 x=0$$

Variable x cannot be equal to 10.

$$x=0$$