Solve 3x(5-2x)=100

$$3 x ( 5 - 2 x ) = 100$$

Solve for x (complex solution)

$$x=\frac{5\sqrt{87}i}{12}+\frac{5}{4}\approx 1.25+3.886407939i$$

$$x=-\frac{5\sqrt{87}i}{12}+\frac{5}{4}\approx 1.25-3.886407939i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 5-2x.

$$15x-6x^{2}=100$$

Subtract 100 from both sides.

$$15x-6x^{2}-100=0$$

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.

$$-6x^{2}+15x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 15 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-6\right)\left(-100\right)}}{2\left(-6\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-6\right)\left(-100\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-15±\sqrt{225+24\left(-100\right)}}{2\left(-6\right)}$$

Multiply 24 times -100.

$$x=\frac{-15±\sqrt{225-2400}}{2\left(-6\right)}$$

Add 225 to -2400.

$$x=\frac{-15±\sqrt{-2175}}{2\left(-6\right)}$$

Take the square root of -2175.

$$x=\frac{-15±5\sqrt{87}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-15±5\sqrt{87}i}{-12}$$

Now solve the equation x=\frac{-15±5\sqrt{87}i}{-12} when ± is plus. Add -15 to 5i\sqrt{87}.

$$x=\frac{-15+5\sqrt{87}i}{-12}$$

Divide -15+5i\sqrt{87} by -12.

$$x=-\frac{5\sqrt{87}i}{12}+\frac{5}{4}$$

Now solve the equation x=\frac{-15±5\sqrt{87}i}{-12} when ± is minus. Subtract 5i\sqrt{87} from -15.

$$x=\frac{-5\sqrt{87}i-15}{-12}$$

Divide -15-5i\sqrt{87} by -12.

$$x=\frac{5\sqrt{87}i}{12}+\frac{5}{4}$$

The equation is now solved.

$$x=-\frac{5\sqrt{87}i}{12}+\frac{5}{4} x=\frac{5\sqrt{87}i}{12}+\frac{5}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 5-2x.

$$15x-6x^{2}=100$$

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.

$$-6x^{2}+15x=100$$

Divide both sides by -6.

$$\frac{-6x^{2}+15x}{-6}=\frac{100}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{15}{-6}x=\frac{100}{-6}$$

Reduce the fraction \frac{15}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{5}{2}x=\frac{100}{-6}$$

Reduce the fraction \frac{100}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{2}x=-\frac{50}{3}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{50}{3}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{50}{3}+\frac{25}{16}$$

Add -\frac{50}{3} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{725}{48}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=-\frac{725}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{725}{48}}$$

Simplify.

$$x-\frac{5}{4}=\frac{5\sqrt{87}i}{12} x-\frac{5}{4}=-\frac{5\sqrt{87}i}{12}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{5\sqrt{87}i}{12}+\frac{5}{4} x=-\frac{5\sqrt{87}i}{12}+\frac{5}{4}$$