Solve x-1=5x-1/3x

$$x-1= \frac{ 5x-1 }{ 3x }$$

Solve for x

$$x = \frac{\sqrt{13} + 4}{3} \approx 2.535183758$$

$$x=\frac{4-\sqrt{13}}{3}\approx 0.131482908$$

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.

$$3xx+3x\left(-1\right)=5x-1$$

Multiply x and x to get x^{2}.

$$3x^{2}+3x\left(-1\right)=5x-1$$

Multiply 3 and -1 to get -3.

$$3x^{2}-3x=5x-1$$

Subtract 5x from both sides.

$$3x^{2}-3x-5x=-1$$

Combine -3x and -5x to get -8x.

$$3x^{2}-8x=-1$$

Add 1 to both sides.

$$3x^{2}-8x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3}}{2\times 3}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-8\right)±\sqrt{64-12}}{2\times 3}$$

Add 64 to -12.

$$x=\frac{-\left(-8\right)±\sqrt{52}}{2\times 3}$$

Take the square root of 52.

$$x=\frac{-\left(-8\right)±2\sqrt{13}}{2\times 3}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{8±2\sqrt{13}}{6}$$

Now solve the equation x=\frac{8±2\sqrt{13}}{6} when ± is plus. Add 8 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+8}{6}$$

Divide 8+2\sqrt{13} by 6.

$$x=\frac{\sqrt{13}+4}{3}$$

Now solve the equation x=\frac{8±2\sqrt{13}}{6} when ± is minus. Subtract 2\sqrt{13} from 8.

$$x=\frac{8-2\sqrt{13}}{6}$$

Divide 8-2\sqrt{13} by 6.

$$x=\frac{4-\sqrt{13}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{13}+4}{3} x=\frac{4-\sqrt{13}}{3}$$

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 3x.

$$3xx+3x\left(-1\right)=5x-1$$

Multiply x and x to get x^{2}.

$$3x^{2}+3x\left(-1\right)=5x-1$$

Multiply 3 and -1 to get -3.

$$3x^{2}-3x=5x-1$$

Subtract 5x from both sides.

$$3x^{2}-3x-5x=-1$$

Combine -3x and -5x to get -8x.

$$3x^{2}-8x=-1$$

Divide both sides by 3.

$$\frac{3x^{2}-8x}{3}=-\frac{1}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{8}{3}x=-\frac{1}{3}$$

Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{1}{3}+\left(-\frac{4}{3}\right)^{2}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{1}{3}+\frac{16}{9}$$

Add -\frac{1}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{13}{9}$$

Factor x^{2}-\frac{8}{3}x+\frac{16}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{13}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{13}{9}}$$

Simplify.

$$x-\frac{4}{3}=\frac{\sqrt{13}}{3} x-\frac{4}{3}=-\frac{\sqrt{13}}{3}$$

Add \frac{4}{3} to both sides of the equation.

$$x=\frac{\sqrt{13}+4}{3} x=\frac{4-\sqrt{13}}{3}$$