Solve 4/7x-8=3x-1

$$\frac{ 4 }{ 7x-8 } =3x-1$$

Solve for x

$$x=\frac{1}{7}\approx 0.142857143$$

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

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{8}{7} since division by zero is not defined. Multiply both sides of the equation by 7x-8.

$$4=3x\left(7x-8\right)+\left(7x-8\right)\left(-1\right)$$

Use the distributive property to multiply 3x by 7x-8.

$$4=21x^{2}-24x+\left(7x-8\right)\left(-1\right)$$

Use the distributive property to multiply 7x-8 by -1.

$$4=21x^{2}-24x-7x+8$$

Combine -24x and -7x to get -31x.

$$4=21x^{2}-31x+8$$

Swap sides so that all variable terms are on the left hand side.

$$21x^{2}-31x+8=4$$

Subtract 4 from both sides.

$$21x^{2}-31x+8-4=0$$

Subtract 4 from 8 to get 4.

$$21x^{2}-31x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -31 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 21\times 4}}{2\times 21}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 21\times 4}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-31\right)±\sqrt{961-84\times 4}}{2\times 21}$$

Multiply -84 times 4.

$$x=\frac{-\left(-31\right)±\sqrt{961-336}}{2\times 21}$$

Add 961 to -336.

$$x=\frac{-\left(-31\right)±\sqrt{625}}{2\times 21}$$

Take the square root of 625.

$$x=\frac{-\left(-31\right)±25}{2\times 21}$$

The opposite of -31 is 31.

$$x=\frac{31±25}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{31±25}{42}$$

Now solve the equation x=\frac{31±25}{42} when ± is plus. Add 31 to 25.

$$x=\frac{56}{42}$$

Reduce the fraction \frac{56}{42} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{31±25}{42} when ± is minus. Subtract 25 from 31.

$$x=\frac{6}{42}$$

Reduce the fraction \frac{6}{42} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{7}$$

The equation is now solved.

$$x=\frac{4}{3} x=\frac{1}{7}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{8}{7} since division by zero is not defined. Multiply both sides of the equation by 7x-8.

$$4=3x\left(7x-8\right)+\left(7x-8\right)\left(-1\right)$$

Use the distributive property to multiply 3x by 7x-8.

$$4=21x^{2}-24x+\left(7x-8\right)\left(-1\right)$$

Use the distributive property to multiply 7x-8 by -1.

$$4=21x^{2}-24x-7x+8$$

Combine -24x and -7x to get -31x.

$$4=21x^{2}-31x+8$$

Swap sides so that all variable terms are on the left hand side.

$$21x^{2}-31x+8=4$$

Subtract 8 from both sides.

$$21x^{2}-31x=4-8$$

Subtract 8 from 4 to get -4.

$$21x^{2}-31x=-4$$

Divide both sides by 21.

$$\frac{21x^{2}-31x}{21}=-\frac{4}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}-\frac{31}{21}x=-\frac{4}{21}$$

Divide -\frac{31}{21}, the coefficient of the x term, by 2 to get -\frac{31}{42}. Then add the square of -\frac{31}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{21}x+\left(-\frac{31}{42}\right)^{2}=-\frac{4}{21}+\left(-\frac{31}{42}\right)^{2}$$

Square -\frac{31}{42} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{21}x+\frac{961}{1764}=-\frac{4}{21}+\frac{961}{1764}$$

Add -\frac{4}{21} to \frac{961}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{21}x+\frac{961}{1764}=\frac{625}{1764}$$

Factor x^{2}-\frac{31}{21}x+\frac{961}{1764}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{42}\right)^{2}=\frac{625}{1764}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{42}\right)^{2}}=\sqrt{\frac{625}{1764}}$$

Simplify.

$$x-\frac{31}{42}=\frac{25}{42} x-\frac{31}{42}=-\frac{25}{42}$$

Add \frac{31}{42} to both sides of the equation.

$$x=\frac{4}{3} x=\frac{1}{7}$$