Solve (50-2x)3x=600

$$(50-2x)3x=600$$

Solve for x

$$x=5$$

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 50-2x by 3.

$$\left(150-6x\right)x=600$$

Use the distributive property to multiply 150-6x by x.

$$150x-6x^{2}=600$$

Subtract 600 from both sides.

$$150x-6x^{2}-600=0$$

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.

$$-6x^{2}+150x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 150 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\left(-6\right)\left(-600\right)}}{2\left(-6\right)}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\left(-6\right)\left(-600\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-150±\sqrt{22500+24\left(-600\right)}}{2\left(-6\right)}$$

Multiply 24 times -600.

$$x=\frac{-150±\sqrt{22500-14400}}{2\left(-6\right)}$$

Add 22500 to -14400.

$$x=\frac{-150±\sqrt{8100}}{2\left(-6\right)}$$

Take the square root of 8100.

$$x=\frac{-150±90}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-150±90}{-12}$$

Now solve the equation x=\frac{-150±90}{-12} when ± is plus. Add -150 to 90.

$$x=-\frac{60}{-12}$$

Divide -60 by -12.

$$x=5$$

Now solve the equation x=\frac{-150±90}{-12} when ± is minus. Subtract 90 from -150.

$$x=-\frac{240}{-12}$$

Divide -240 by -12.

$$x=20$$

The equation is now solved.

$$x=5 x=20$$

Steps for Completing the Square

Use the distributive property to multiply 50-2x by 3.

$$\left(150-6x\right)x=600$$

Use the distributive property to multiply 150-6x by x.

$$150x-6x^{2}=600$$

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.

$$-6x^{2}+150x=600$$

Divide both sides by -6.

$$\frac{-6x^{2}+150x}{-6}=\frac{600}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{150}{-6}x=\frac{600}{-6}$$

Divide 150 by -6.

$$x^{2}-25x=\frac{600}{-6}$$

Divide 600 by -6.

$$x^{2}-25x=-100$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-100+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-100+\frac{625}{4}$$

Add -100 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{225}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{15}{2} x-\frac{25}{2}=-\frac{15}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=20 x=5$$