Solve 10-2x/x*3x=2.4

$$\frac{ 10-2x }{ x \times 3x } = 2.4$$

Solve for x

$$x = \frac{5 \sqrt{73} - 5}{36} \approx 1.047778298$$

$$x=\frac{-5\sqrt{73}-5}{36}\approx -1.325556076$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x^{2}.

$$10-2x=7.2x^{2}$$

Subtract 7.2x^{2} from both sides.

$$10-2x-7.2x^{2}=0$$

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.

$$-7.2x^{2}-2x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7.2 for a, -2 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-7.2\right)\times 10}}{2\left(-7.2\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-7.2\right)\times 10}}{2\left(-7.2\right)}$$

Multiply -4 times -7.2.

$$x=\frac{-\left(-2\right)±\sqrt{4+28.8\times 10}}{2\left(-7.2\right)}$$

Multiply 28.8 times 10.

$$x=\frac{-\left(-2\right)±\sqrt{4+288}}{2\left(-7.2\right)}$$

Add 4 to 288.

$$x=\frac{-\left(-2\right)±\sqrt{292}}{2\left(-7.2\right)}$$

Take the square root of 292.

$$x=\frac{-\left(-2\right)±2\sqrt{73}}{2\left(-7.2\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{73}}{2\left(-7.2\right)}$$

Multiply 2 times -7.2.

$$x=\frac{2±2\sqrt{73}}{-14.4}$$

Now solve the equation x=\frac{2±2\sqrt{73}}{-14.4} when ± is plus. Add 2 to 2\sqrt{73}.

$$x=\frac{2\sqrt{73}+2}{-14.4}$$

Divide 2+2\sqrt{73} by -14.4 by multiplying 2+2\sqrt{73} by the reciprocal of -14.4.

$$x=\frac{-5\sqrt{73}-5}{36}$$

Now solve the equation x=\frac{2±2\sqrt{73}}{-14.4} when ± is minus. Subtract 2\sqrt{73} from 2.

$$x=\frac{2-2\sqrt{73}}{-14.4}$$

Divide 2-2\sqrt{73} by -14.4 by multiplying 2-2\sqrt{73} by the reciprocal of -14.4.

$$x=\frac{5\sqrt{73}-5}{36}$$

The equation is now solved.

$$x=\frac{-5\sqrt{73}-5}{36} x=\frac{5\sqrt{73}-5}{36}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x^{2}.

$$10-2x=7.2x^{2}$$

Subtract 7.2x^{2} from both sides.

$$10-2x-7.2x^{2}=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$-2x-7.2x^{2}=-10$$

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.

$$-7.2x^{2}-2x=-10$$

Divide both sides of the equation by -7.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-7.2x^{2}-2x}{-7.2}=-\frac{10}{-7.2}$$

Dividing by -7.2 undoes the multiplication by -7.2.

$$x^{2}+\left(-\frac{2}{-7.2}\right)x=-\frac{10}{-7.2}$$

Divide -2 by -7.2 by multiplying -2 by the reciprocal of -7.2.

$$x^{2}+\frac{5}{18}x=-\frac{10}{-7.2}$$

Divide -10 by -7.2 by multiplying -10 by the reciprocal of -7.2.

$$x^{2}+\frac{5}{18}x=\frac{25}{18}$$

Divide \frac{5}{18}, the coefficient of the x term, by 2 to get \frac{5}{36}. Then add the square of \frac{5}{36} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{18}x+\frac{5}{36}^{2}=\frac{25}{18}+\frac{5}{36}^{2}$$

Square \frac{5}{36} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{25}{18}+\frac{25}{1296}$$

Add \frac{25}{18} to \frac{25}{1296} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{1825}{1296}$$

Factor x^{2}+\frac{5}{18}x+\frac{25}{1296}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{36}\right)^{2}=\frac{1825}{1296}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{36}\right)^{2}}=\sqrt{\frac{1825}{1296}}$$

Simplify.

$$x+\frac{5}{36}=\frac{5\sqrt{73}}{36} x+\frac{5}{36}=-\frac{5\sqrt{73}}{36}$$

Subtract \frac{5}{36} from both sides of the equation.

$$x=\frac{5\sqrt{73}-5}{36} x=\frac{-5\sqrt{73}-5}{36}$$