Solve 75x-25x^2=(15-5x)(24-5x)

$$75 x - 25 x ^ { 2 } = ( 15 - 5 x ) ( 24 - 5 x )$$

Solve for x

$$x=3$$

$$x = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 15-5x by 24-5x and combine like terms.

$$75x-25x^{2}=360-195x+25x^{2}$$

Subtract 360 from both sides.

$$75x-25x^{2}-360=-195x+25x^{2}$$

Add 195x to both sides.

$$75x-25x^{2}-360+195x=25x^{2}$$

Combine 75x and 195x to get 270x.

$$270x-25x^{2}-360=25x^{2}$$

Subtract 25x^{2} from both sides.

$$270x-25x^{2}-360-25x^{2}=0$$

Combine -25x^{2} and -25x^{2} to get -50x^{2}.

$$270x-50x^{2}-360=0$$

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.

$$-50x^{2}+270x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 270 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-270±\sqrt{270^{2}-4\left(-50\right)\left(-360\right)}}{2\left(-50\right)}$$

Square 270.

$$x=\frac{-270±\sqrt{72900-4\left(-50\right)\left(-360\right)}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-270±\sqrt{72900+200\left(-360\right)}}{2\left(-50\right)}$$

Multiply 200 times -360.

$$x=\frac{-270±\sqrt{72900-72000}}{2\left(-50\right)}$$

Add 72900 to -72000.

$$x=\frac{-270±\sqrt{900}}{2\left(-50\right)}$$

Take the square root of 900.

$$x=\frac{-270±30}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=\frac{-270±30}{-100}$$

Now solve the equation x=\frac{-270±30}{-100} when ± is plus. Add -270 to 30.

$$x=-\frac{240}{-100}$$

Reduce the fraction \frac{-240}{-100} to lowest terms by extracting and canceling out 20.

$$x=\frac{12}{5}$$

Now solve the equation x=\frac{-270±30}{-100} when ± is minus. Subtract 30 from -270.

$$x=-\frac{300}{-100}$$

Divide -300 by -100.

$$x=3$$

The equation is now solved.

$$x=\frac{12}{5} x=3$$

Steps for Completing the Square

Use the distributive property to multiply 15-5x by 24-5x and combine like terms.

$$75x-25x^{2}=360-195x+25x^{2}$$

Add 195x to both sides.

$$75x-25x^{2}+195x=360+25x^{2}$$

Combine 75x and 195x to get 270x.

$$270x-25x^{2}=360+25x^{2}$$

Subtract 25x^{2} from both sides.

$$270x-25x^{2}-25x^{2}=360$$

Combine -25x^{2} and -25x^{2} to get -50x^{2}.

$$270x-50x^{2}=360$$

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.

$$-50x^{2}+270x=360$$

Divide both sides by -50.

$$\frac{-50x^{2}+270x}{-50}=\frac{360}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{270}{-50}x=\frac{360}{-50}$$

Reduce the fraction \frac{270}{-50} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{27}{5}x=\frac{360}{-50}$$

Reduce the fraction \frac{360}{-50} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{27}{5}x=-\frac{36}{5}$$

Divide -\frac{27}{5}, the coefficient of the x term, by 2 to get -\frac{27}{10}. Then add the square of -\frac{27}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{27}{5}x+\left(-\frac{27}{10}\right)^{2}=-\frac{36}{5}+\left(-\frac{27}{10}\right)^{2}$$

Square -\frac{27}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{27}{5}x+\frac{729}{100}=-\frac{36}{5}+\frac{729}{100}$$

Add -\frac{36}{5} to \frac{729}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{27}{5}x+\frac{729}{100}=\frac{9}{100}$$

Factor x^{2}-\frac{27}{5}x+\frac{729}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{27}{10}\right)^{2}=\frac{9}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{10}\right)^{2}}=\sqrt{\frac{9}{100}}$$

Simplify.

$$x-\frac{27}{10}=\frac{3}{10} x-\frac{27}{10}=-\frac{3}{10}$$

Add \frac{27}{10} to both sides of the equation.

$$x=3 x=\frac{12}{5}$$