Solve 4/7x-8=3x

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

Solve for x

$$x=\frac{2\sqrt{57}}{21}+\frac{4}{7}\approx 1.290460422$$

$$x=-\frac{2\sqrt{57}}{21}+\frac{4}{7}\approx -0.14760328$$

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.

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

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

$$4=21x^{2}-24x$$

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

$$21x^{2}-24x=4$$

Subtract 4 from both sides.

$$21x^{2}-24x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -24 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 21\left(-4\right)}}{2\times 21}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 21\left(-4\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-24\right)±\sqrt{576-84\left(-4\right)}}{2\times 21}$$

Multiply -84 times -4.

$$x=\frac{-\left(-24\right)±\sqrt{576+336}}{2\times 21}$$

Add 576 to 336.

$$x=\frac{-\left(-24\right)±\sqrt{912}}{2\times 21}$$

Take the square root of 912.

$$x=\frac{-\left(-24\right)±4\sqrt{57}}{2\times 21}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{57}}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{24±4\sqrt{57}}{42}$$

Now solve the equation x=\frac{24±4\sqrt{57}}{42} when ± is plus. Add 24 to 4\sqrt{57}.

$$x=\frac{4\sqrt{57}+24}{42}$$

Divide 24+4\sqrt{57} by 42.

$$x=\frac{2\sqrt{57}}{21}+\frac{4}{7}$$

Now solve the equation x=\frac{24±4\sqrt{57}}{42} when ± is minus. Subtract 4\sqrt{57} from 24.

$$x=\frac{24-4\sqrt{57}}{42}$$

Divide 24-4\sqrt{57} by 42.

$$x=-\frac{2\sqrt{57}}{21}+\frac{4}{7}$$

The equation is now solved.

$$x=\frac{2\sqrt{57}}{21}+\frac{4}{7} x=-\frac{2\sqrt{57}}{21}+\frac{4}{7}$$

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.

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

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

$$4=21x^{2}-24x$$

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

$$21x^{2}-24x=4$$

Divide both sides by 21.

$$\frac{21x^{2}-24x}{21}=\frac{4}{21}$$

Dividing by 21 undoes the multiplication by 21.

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

Reduce the fraction \frac{-24}{21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{8}{7}x=\frac{4}{21}$$

Divide -\frac{8}{7}, the coefficient of the x term, by 2 to get -\frac{4}{7}. Then add the square of -\frac{4}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{4}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{4}{21}+\frac{16}{49}$$

Add \frac{4}{21} to \frac{16}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{76}{147}$$

Factor x^{2}-\frac{8}{7}x+\frac{16}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{7}\right)^{2}=\frac{76}{147}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{7}\right)^{2}}=\sqrt{\frac{76}{147}}$$

Simplify.

$$x-\frac{4}{7}=\frac{2\sqrt{57}}{21} x-\frac{4}{7}=-\frac{2\sqrt{57}}{21}$$

Add \frac{4}{7} to both sides of the equation.

$$x=\frac{2\sqrt{57}}{21}+\frac{4}{7} x=-\frac{2\sqrt{57}}{21}+\frac{4}{7}$$