Solve 3x-5/4=3/x

$$\frac{ 3x-5 }{ 4 } = \frac{ 3 }{ x }$$

Solve for x

$$x = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

$$x=3$$

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 4x, the least common multiple of 4,x.

$$x\left(3x-5\right)=4\times 3$$

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

$$3x^{2}-5x=4\times 3$$

Multiply 4 and 3 to get 12.

$$3x^{2}-5x=12$$

Subtract 12 from both sides.

$$3x^{2}-5x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\times 3}$$

Add 25 to 144.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\times 3}$$

Take the square root of 169.

$$x=\frac{-\left(-5\right)±13}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±13}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±13}{6}$$

Now solve the equation x=\frac{5±13}{6} when ± is plus. Add 5 to 13.

$$x=\frac{18}{6}$$

Divide 18 by 6.

$$x=3$$

Now solve the equation x=\frac{5±13}{6} when ± is minus. Subtract 13 from 5.

$$x=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{4}{3}$$

The equation is now solved.

$$x=3 x=-\frac{4}{3}$$

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 4x, the least common multiple of 4,x.

$$x\left(3x-5\right)=4\times 3$$

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

$$3x^{2}-5x=4\times 3$$

Multiply 4 and 3 to get 12.

$$3x^{2}-5x=12$$

Divide both sides by 3.

$$\frac{3x^{2}-5x}{3}=\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{5}{3}x=\frac{12}{3}$$

Divide 12 by 3.

$$x^{2}-\frac{5}{3}x=4$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=4+\left(-\frac{5}{6}\right)^{2}$$

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

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=4+\frac{25}{36}$$

Add 4 to \frac{25}{36}.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{169}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{169}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{13}{6} x-\frac{5}{6}=-\frac{13}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$x=3 x=-\frac{4}{3}$$