Solve 26/xdiv0.6x=26/x+3

$$\frac { 26 } { x } \div 0.6 x = \frac { 26 } { x + 3 }$$

Solve for x

$$x=-2.4$$

Steps Using Factoring

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$\left(\frac{5}{3}x+5\right)\times \frac{26}{x}x=26$$

Express \frac{26}{x}x as a single fraction.

$$\left(\frac{5}{3}x+5\right)\times \frac{26x}{x}=26$$

Use the distributive property to multiply \frac{5}{3}x+5 by \frac{26x}{x}.

$$\frac{5}{3}x\times \frac{26x}{x}+5\times \frac{26x}{x}=26$$

Multiply \frac{5}{3} times \frac{26x}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 26x}{3x}x+5\times \frac{26x}{x}=26$$

Express 5\times \frac{26x}{x} as a single fraction.

$$\frac{5\times 26x}{3x}x+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130x}{3x}x+\frac{5\times 26x}{x}=26$$

Express \frac{130x}{3x}x as a single fraction.

$$\frac{130xx}{3x}+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130xx}{3x}+\frac{130x}{x}=26$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x and x is 3x. Multiply \frac{130x}{x} times \frac{3}{3}.

$$\frac{130xx}{3x}+\frac{3\times 130x}{3x}=26$$

Since \frac{130xx}{3x} and \frac{3\times 130x}{3x} have the same denominator, add them by adding their numerators.

$$\frac{130xx+3\times 130x}{3x}=26$$

Do the multiplications in 130xx+3\times 130x.

$$\frac{130x^{2}+390x}{3x}=26$$

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

$$130x^{2}+390x=78x$$

Subtract 78x from both sides.

$$130x^{2}+390x-78x=0$$

Combine 390x and -78x to get 312x.

$$130x^{2}+312x=0$$

Factor out x.

$$x\left(130x+312\right)=0$$

To find equation solutions, solve x=0 and 130x+312=0.

$$x=0 x=-\frac{12}{5}$$

Variable x cannot be equal to 0.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$\left(\frac{5}{3}x+5\right)\times \frac{26}{x}x=26$$

Express \frac{26}{x}x as a single fraction.

$$\left(\frac{5}{3}x+5\right)\times \frac{26x}{x}=26$$

Use the distributive property to multiply \frac{5}{3}x+5 by \frac{26x}{x}.

$$\frac{5}{3}x\times \frac{26x}{x}+5\times \frac{26x}{x}=26$$

Multiply \frac{5}{3} times \frac{26x}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 26x}{3x}x+5\times \frac{26x}{x}=26$$

Express 5\times \frac{26x}{x} as a single fraction.

$$\frac{5\times 26x}{3x}x+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130x}{3x}x+\frac{5\times 26x}{x}=26$$

Express \frac{130x}{3x}x as a single fraction.

$$\frac{130xx}{3x}+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130xx}{3x}+\frac{130x}{x}=26$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x and x is 3x. Multiply \frac{130x}{x} times \frac{3}{3}.

$$\frac{130xx}{3x}+\frac{3\times 130x}{3x}=26$$

Since \frac{130xx}{3x} and \frac{3\times 130x}{3x} have the same denominator, add them by adding their numerators.

$$\frac{130xx+3\times 130x}{3x}=26$$

Do the multiplications in 130xx+3\times 130x.

$$\frac{130x^{2}+390x}{3x}=26$$

Subtract 26 from both sides.

$$\frac{130x^{2}+390x}{3x}-26=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 26 times \frac{3x}{3x}.

$$\frac{130x^{2}+390x}{3x}-\frac{26\times 3x}{3x}=0$$

Since \frac{130x^{2}+390x}{3x} and \frac{26\times 3x}{3x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{130x^{2}+390x-26\times 3x}{3x}=0$$

Do the multiplications in 130x^{2}+390x-26\times 3x.

$$\frac{130x^{2}+390x-78x}{3x}=0$$

Combine like terms in 130x^{2}+390x-78x.

$$\frac{130x^{2}+312x}{3x}=0$$

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

$$130x^{2}+312x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 130 for a, 312 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-312±\sqrt{312^{2}}}{2\times 130}$$

Take the square root of 312^{2}.

$$x=\frac{-312±312}{2\times 130}$$

Multiply 2 times 130.

$$x=\frac{-312±312}{260}$$

Now solve the equation x=\frac{-312±312}{260} when ± is plus. Add -312 to 312.

$$x=\frac{0}{260}$$

Divide 0 by 260.

$$x=0$$

Now solve the equation x=\frac{-312±312}{260} when ± is minus. Subtract 312 from -312.

$$x=-\frac{624}{260}$$

Reduce the fraction \frac{-624}{260} to lowest terms by extracting and canceling out 52.

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

The equation is now solved.

$$x=0 x=-\frac{12}{5}$$

Variable x cannot be equal to 0.

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

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$\left(\frac{5}{3}x+5\right)\times \frac{26}{x}x=26$$

Express \frac{26}{x}x as a single fraction.

$$\left(\frac{5}{3}x+5\right)\times \frac{26x}{x}=26$$

Use the distributive property to multiply \frac{5}{3}x+5 by \frac{26x}{x}.

$$\frac{5}{3}x\times \frac{26x}{x}+5\times \frac{26x}{x}=26$$

Multiply \frac{5}{3} times \frac{26x}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 26x}{3x}x+5\times \frac{26x}{x}=26$$

Express 5\times \frac{26x}{x} as a single fraction.

$$\frac{5\times 26x}{3x}x+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130x}{3x}x+\frac{5\times 26x}{x}=26$$

Express \frac{130x}{3x}x as a single fraction.

$$\frac{130xx}{3x}+\frac{5\times 26x}{x}=26$$

Multiply 5 and 26 to get 130.

$$\frac{130xx}{3x}+\frac{130x}{x}=26$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x and x is 3x. Multiply \frac{130x}{x} times \frac{3}{3}.

$$\frac{130xx}{3x}+\frac{3\times 130x}{3x}=26$$

Since \frac{130xx}{3x} and \frac{3\times 130x}{3x} have the same denominator, add them by adding their numerators.

$$\frac{130xx+3\times 130x}{3x}=26$$

Do the multiplications in 130xx+3\times 130x.

$$\frac{130x^{2}+390x}{3x}=26$$

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

$$130x^{2}+390x=78x$$

Subtract 78x from both sides.

$$130x^{2}+390x-78x=0$$

Combine 390x and -78x to get 312x.

$$130x^{2}+312x=0$$

Divide both sides by 130.

$$\frac{130x^{2}+312x}{130}=\frac{0}{130}$$

Dividing by 130 undoes the multiplication by 130.

$$x^{2}+\frac{312}{130}x=\frac{0}{130}$$

Reduce the fraction \frac{312}{130} to lowest terms by extracting and canceling out 26.

$$x^{2}+\frac{12}{5}x=\frac{0}{130}$$

Divide 0 by 130.

$$x^{2}+\frac{12}{5}x=0$$

Divide \frac{12}{5}, the coefficient of the x term, by 2 to get \frac{6}{5}. Then add the square of \frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=\left(\frac{6}{5}\right)^{2}$$

Square \frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{36}{25}$$

Factor x^{2}+\frac{12}{5}x+\frac{36}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{5}\right)^{2}=\frac{36}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Simplify.

$$x+\frac{6}{5}=\frac{6}{5} x+\frac{6}{5}=-\frac{6}{5}$$

Subtract \frac{6}{5} from both sides of the equation.

$$x=0 x=-\frac{12}{5}$$

Variable x cannot be equal to 0.

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