Solve 3x-4=-1/3x+16/3

$$3 x - 4 = \frac { - 1 } { 3 x } + \frac { 16 } { 3 }$$

Solve for x

$$x=\frac{14-\sqrt{187}}{9}\approx 0.036133963$$

$$x = \frac{\sqrt{187} + 14}{9} \approx 3.074977148$$

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, the least common multiple of 3x,3.

$$3x\times 3x+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

Multiply 3 and 3 to get 9.

$$9xx+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

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

$$9x^{2}+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

Multiply 3 and -4 to get -12.

$$9x^{2}-12x=-1+3x\times \frac{16}{3}$$

Multiply 3 and \frac{16}{3} to get 16.

$$9x^{2}-12x=-1+16x$$

Subtract -1 from both sides.

$$9x^{2}-12x-\left(-1\right)=16x$$

The opposite of -1 is 1.

$$9x^{2}-12x+1=16x$$

Subtract 16x from both sides.

$$9x^{2}-12x+1-16x=0$$

Combine -12x and -16x to get -28x.

$$9x^{2}-28x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -28 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 9}}{2\times 9}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 9}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-28\right)±\sqrt{784-36}}{2\times 9}$$

Add 784 to -36.

$$x=\frac{-\left(-28\right)±\sqrt{748}}{2\times 9}$$

Take the square root of 748.

$$x=\frac{-\left(-28\right)±2\sqrt{187}}{2\times 9}$$

The opposite of -28 is 28.

$$x=\frac{28±2\sqrt{187}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{28±2\sqrt{187}}{18}$$

Now solve the equation x=\frac{28±2\sqrt{187}}{18} when ± is plus. Add 28 to 2\sqrt{187}.

$$x=\frac{2\sqrt{187}+28}{18}$$

Divide 28+2\sqrt{187} by 18.

$$x=\frac{\sqrt{187}+14}{9}$$

Now solve the equation x=\frac{28±2\sqrt{187}}{18} when ± is minus. Subtract 2\sqrt{187} from 28.

$$x=\frac{28-2\sqrt{187}}{18}$$

Divide 28-2\sqrt{187} by 18.

$$x=\frac{14-\sqrt{187}}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{187}+14}{9} x=\frac{14-\sqrt{187}}{9}$$

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, the least common multiple of 3x,3.

$$3x\times 3x+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

Multiply 3 and 3 to get 9.

$$9xx+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

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

$$9x^{2}+3x\left(-4\right)=-1+3x\times \frac{16}{3}$$

Multiply 3 and -4 to get -12.

$$9x^{2}-12x=-1+3x\times \frac{16}{3}$$

Multiply 3 and \frac{16}{3} to get 16.

$$9x^{2}-12x=-1+16x$$

Subtract 16x from both sides.

$$9x^{2}-12x-16x=-1$$

Combine -12x and -16x to get -28x.

$$9x^{2}-28x=-1$$

Divide both sides by 9.

$$\frac{9x^{2}-28x}{9}=-\frac{1}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{28}{9}x=-\frac{1}{9}$$

Divide -\frac{28}{9}, the coefficient of the x term, by 2 to get -\frac{14}{9}. Then add the square of -\frac{14}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{9}x+\left(-\frac{14}{9}\right)^{2}=-\frac{1}{9}+\left(-\frac{14}{9}\right)^{2}$$

Square -\frac{14}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{9}x+\frac{196}{81}=-\frac{1}{9}+\frac{196}{81}$$

Add -\frac{1}{9} to \frac{196}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{28}{9}x+\frac{196}{81}=\frac{187}{81}$$

Factor x^{2}-\frac{28}{9}x+\frac{196}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{9}\right)^{2}=\frac{187}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{9}\right)^{2}}=\sqrt{\frac{187}{81}}$$

Simplify.

$$x-\frac{14}{9}=\frac{\sqrt{187}}{9} x-\frac{14}{9}=-\frac{\sqrt{187}}{9}$$

Add \frac{14}{9} to both sides of the equation.

$$x=\frac{\sqrt{187}+14}{9} x=\frac{14-\sqrt{187}}{9}$$