Solve 360x=210x+15x^2

$$360 x = 210 x + 15 x ^ { 2 }$$

Solve for x

$$x=10$$

$$x=0$$

Steps Using Factoring

Subtract 210x from both sides.

$$360x-210x=15x^{2}$$

Combine 360x and -210x to get 150x.

$$150x=15x^{2}$$

Subtract 15x^{2} from both sides.

$$150x-15x^{2}=0$$

Factor out x.

$$x\left(150-15x\right)=0$$

To find equation solutions, solve x=0 and 150-15x=0.

$$x=0 x=10$$

Steps Using the Quadratic Formula

Subtract 210x from both sides.

$$360x-210x=15x^{2}$$

Combine 360x and -210x to get 150x.

$$150x=15x^{2}$$

Subtract 15x^{2} from both sides.

$$150x-15x^{2}=0$$

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.

$$-15x^{2}+150x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 150 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}}}{2\left(-15\right)}$$

Take the square root of 150^{2}.

$$x=\frac{-150±150}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-150±150}{-30}$$

Now solve the equation x=\frac{-150±150}{-30} when ± is plus. Add -150 to 150.

$$x=\frac{0}{-30}$$

Divide 0 by -30.

$$x=0$$

Now solve the equation x=\frac{-150±150}{-30} when ± is minus. Subtract 150 from -150.

$$x=-\frac{300}{-30}$$

Divide -300 by -30.

$$x=10$$

The equation is now solved.

$$x=0 x=10$$

Steps for Completing the Square

Subtract 210x from both sides.

$$360x-210x=15x^{2}$$

Combine 360x and -210x to get 150x.

$$150x=15x^{2}$$

Subtract 15x^{2} from both sides.

$$150x-15x^{2}=0$$

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.

$$-15x^{2}+150x=0$$

Divide both sides by -15.

$$\frac{-15x^{2}+150x}{-15}=\frac{0}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{150}{-15}x=\frac{0}{-15}$$

Divide 150 by -15.

$$x^{2}-10x=\frac{0}{-15}$$

Divide 0 by -15.

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

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.

$$x=10 x=0$$