Solve x(15-2x/2)(0.10)=1

$$x( \frac{ 15-2x }{ 2 } )(0.10)=1$$

Solve for x

$$x = \frac{\sqrt{65} + 15}{4} \approx 5.765564437$$

$$x = \frac{15 - \sqrt{65}}{4} \approx 1.734435563$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(15-2x\right)\times 0.1=2$$

Use the distributive property to multiply x by 15-2x.

$$\left(15x-2x^{2}\right)\times 0.1=2$$

Use the distributive property to multiply 15x-2x^{2} by 0.1.

$$1.5x-0.2x^{2}=2$$

Subtract 2 from both sides.

$$1.5x-0.2x^{2}-2=0$$

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.

$$-0.2x^{2}+1.5x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.2 for a, 1.5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.5±\sqrt{1.5^{2}-4\left(-0.2\right)\left(-2\right)}}{2\left(-0.2\right)}$$

Square 1.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.5±\sqrt{2.25-4\left(-0.2\right)\left(-2\right)}}{2\left(-0.2\right)}$$

Multiply -4 times -0.2.

$$x=\frac{-1.5±\sqrt{2.25+0.8\left(-2\right)}}{2\left(-0.2\right)}$$

Multiply 0.8 times -2.

$$x=\frac{-1.5±\sqrt{2.25-1.6}}{2\left(-0.2\right)}$$

Add 2.25 to -1.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.5±\sqrt{0.65}}{2\left(-0.2\right)}$$

Take the square root of 0.65.

$$x=\frac{-1.5±\frac{\sqrt{65}}{10}}{2\left(-0.2\right)}$$

Multiply 2 times -0.2.

$$x=\frac{-1.5±\frac{\sqrt{65}}{10}}{-0.4}$$

Now solve the equation x=\frac{-1.5±\frac{\sqrt{65}}{10}}{-0.4} when ± is plus. Add -1.5 to \frac{\sqrt{65}}{10}.

$$x=\frac{\frac{\sqrt{65}}{10}-\frac{3}{2}}{-0.4}$$

Divide -\frac{3}{2}+\frac{\sqrt{65}}{10} by -0.4 by multiplying -\frac{3}{2}+\frac{\sqrt{65}}{10} by the reciprocal of -0.4.

$$x=\frac{15-\sqrt{65}}{4}$$

Now solve the equation x=\frac{-1.5±\frac{\sqrt{65}}{10}}{-0.4} when ± is minus. Subtract \frac{\sqrt{65}}{10} from -1.5.

$$x=\frac{-\frac{\sqrt{65}}{10}-\frac{3}{2}}{-0.4}$$

Divide -\frac{3}{2}-\frac{\sqrt{65}}{10} by -0.4 by multiplying -\frac{3}{2}-\frac{\sqrt{65}}{10} by the reciprocal of -0.4.

$$x=\frac{\sqrt{65}+15}{4}$$

The equation is now solved.

$$x=\frac{15-\sqrt{65}}{4} x=\frac{\sqrt{65}+15}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(15-2x\right)\times 0.1=2$$

Use the distributive property to multiply x by 15-2x.

$$\left(15x-2x^{2}\right)\times 0.1=2$$

Use the distributive property to multiply 15x-2x^{2} by 0.1.

$$1.5x-0.2x^{2}=2$$

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.

$$-0.2x^{2}+1.5x=2$$

Multiply both sides by -5.

$$\frac{-0.2x^{2}+1.5x}{-0.2}=\frac{2}{-0.2}$$

Dividing by -0.2 undoes the multiplication by -0.2.

$$x^{2}+\frac{1.5}{-0.2}x=\frac{2}{-0.2}$$

Divide 1.5 by -0.2 by multiplying 1.5 by the reciprocal of -0.2.

$$x^{2}-7.5x=\frac{2}{-0.2}$$

Divide 2 by -0.2 by multiplying 2 by the reciprocal of -0.2.

$$x^{2}-7.5x=-10$$

Divide -7.5, the coefficient of the x term, by 2 to get -3.75. Then add the square of -3.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7.5x+\left(-3.75\right)^{2}=-10+\left(-3.75\right)^{2}$$

Square -3.75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7.5x+14.0625=-10+14.0625$$

Add -10 to 14.0625.

$$x^{2}-7.5x+14.0625=4.0625$$

Factor x^{2}-7.5x+14.0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3.75\right)^{2}=4.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3.75\right)^{2}}=\sqrt{4.0625}$$

Simplify.

$$x-3.75=\frac{\sqrt{65}}{4} x-3.75=-\frac{\sqrt{65}}{4}$$

Add 3.75 to both sides of the equation.

$$x=\frac{\sqrt{65}+15}{4} x=\frac{15-\sqrt{65}}{4}$$