Solve 15x*(25-x)=2500

$$15 x \times ( 25 - x ) = 2500$$

Solve for x (complex solution)

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}\approx 12.5+3.227486122i$$

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}\approx 12.5-3.227486122i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 15x by 25-x.

$$375x-15x^{2}=2500$$

Subtract 2500 from both sides.

$$375x-15x^{2}-2500=0$$

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.

$$-15x^{2}+375x-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 375 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-375±\sqrt{375^{2}-4\left(-15\right)\left(-2500\right)}}{2\left(-15\right)}$$

Square 375.

$$x=\frac{-375±\sqrt{140625-4\left(-15\right)\left(-2500\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-375±\sqrt{140625+60\left(-2500\right)}}{2\left(-15\right)}$$

Multiply 60 times -2500.

$$x=\frac{-375±\sqrt{140625-150000}}{2\left(-15\right)}$$

Add 140625 to -150000.

$$x=\frac{-375±\sqrt{-9375}}{2\left(-15\right)}$$

Take the square root of -9375.

$$x=\frac{-375±25\sqrt{15}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-375±25\sqrt{15}i}{-30}$$

Now solve the equation x=\frac{-375±25\sqrt{15}i}{-30} when ± is plus. Add -375 to 25i\sqrt{15}.

$$x=\frac{-375+25\sqrt{15}i}{-30}$$

Divide -375+25i\sqrt{15} by -30.

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

Now solve the equation x=\frac{-375±25\sqrt{15}i}{-30} when ± is minus. Subtract 25i\sqrt{15} from -375.

$$x=\frac{-25\sqrt{15}i-375}{-30}$$

Divide -375-25i\sqrt{15} by -30.

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

The equation is now solved.

$$x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2} x=\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 15x by 25-x.

$$375x-15x^{2}=2500$$

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.

$$-15x^{2}+375x=2500$$

Divide both sides by -15.

$$\frac{-15x^{2}+375x}{-15}=\frac{2500}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{375}{-15}x=\frac{2500}{-15}$$

Divide 375 by -15.

$$x^{2}-25x=\frac{2500}{-15}$$

Reduce the fraction \frac{2500}{-15} to lowest terms by extracting and canceling out 5.

$$x^{2}-25x=-\frac{500}{3}$$

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.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-\frac{500}{3}+\left(-\frac{25}{2}\right)^{2}$$

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

$$x^{2}-25x+\frac{625}{4}=-\frac{500}{3}+\frac{625}{4}$$

Add -\frac{500}{3} to \frac{625}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-25x+\frac{625}{4}=-\frac{125}{12}$$

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}.

$$\left(x-\frac{25}{2}\right)^{2}=-\frac{125}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{-\frac{125}{12}}$$

Simplify.

$$x-\frac{25}{2}=\frac{5\sqrt{15}i}{6} x-\frac{25}{2}=-\frac{5\sqrt{15}i}{6}$$

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

$$x=\frac{5\sqrt{15}i}{6}+\frac{25}{2} x=-\frac{5\sqrt{15}i}{6}+\frac{25}{2}$$