Solve 56/(-2x)=8/3-5x

$$56 / ( - 2 x ) = 8 / 3 - 5 x$$

Solve for x

$$x = \frac{2 \sqrt{319} + 4}{15} \approx 2.648076147$$

$$x=\frac{4-2\sqrt{319}}{15}\approx -2.114742813$$

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 6x, the least common multiple of -2x,3.

$$-3\times 56=6x\times \frac{8}{3}-5x\times 6x$$

Multiply -3 and 56 to get -168.

$$-168=6x\times \frac{8}{3}-5x\times 6x$$

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

$$-168=6x\times \frac{8}{3}-5x^{2}\times 6$$

Multiply 6 and \frac{8}{3} to get 16.

$$-168=16x-5x^{2}\times 6$$

Multiply -5 and 6 to get -30.

$$-168=16x-30x^{2}$$

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

$$16x-30x^{2}=-168$$

Add 168 to both sides.

$$16x-30x^{2}+168=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-30x^{2}+16x+168=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, 16 for b, and 168 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-30\right)\times 168}}{2\left(-30\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-30\right)\times 168}}{2\left(-30\right)}$$

Multiply -4 times -30.

$$x=\frac{-16±\sqrt{256+120\times 168}}{2\left(-30\right)}$$

Multiply 120 times 168.

$$x=\frac{-16±\sqrt{256+20160}}{2\left(-30\right)}$$

Add 256 to 20160.

$$x=\frac{-16±\sqrt{20416}}{2\left(-30\right)}$$

Take the square root of 20416.

$$x=\frac{-16±8\sqrt{319}}{2\left(-30\right)}$$

Multiply 2 times -30.

$$x=\frac{-16±8\sqrt{319}}{-60}$$

Now solve the equation x=\frac{-16±8\sqrt{319}}{-60} when ± is plus. Add -16 to 8\sqrt{319}.

$$x=\frac{8\sqrt{319}-16}{-60}$$

Divide -16+8\sqrt{319} by -60.

$$x=\frac{4-2\sqrt{319}}{15}$$

Now solve the equation x=\frac{-16±8\sqrt{319}}{-60} when ± is minus. Subtract 8\sqrt{319} from -16.

$$x=\frac{-8\sqrt{319}-16}{-60}$$

Divide -16-8\sqrt{319} by -60.

$$x=\frac{2\sqrt{319}+4}{15}$$

The equation is now solved.

$$x=\frac{4-2\sqrt{319}}{15} x=\frac{2\sqrt{319}+4}{15}$$

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 6x, the least common multiple of -2x,3.

$$-3\times 56=6x\times \frac{8}{3}-5x\times 6x$$

Multiply -3 and 56 to get -168.

$$-168=6x\times \frac{8}{3}-5x\times 6x$$

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

$$-168=6x\times \frac{8}{3}-5x^{2}\times 6$$

Multiply 6 and \frac{8}{3} to get 16.

$$-168=16x-5x^{2}\times 6$$

Multiply -5 and 6 to get -30.

$$-168=16x-30x^{2}$$

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

$$16x-30x^{2}=-168$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-30x^{2}+16x=-168$$

Divide both sides by -30.

$$\frac{-30x^{2}+16x}{-30}=-\frac{168}{-30}$$

Dividing by -30 undoes the multiplication by -30.

$$x^{2}+\frac{16}{-30}x=-\frac{168}{-30}$$

Reduce the fraction \frac{16}{-30} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{15}x=-\frac{168}{-30}$$

Reduce the fraction \frac{-168}{-30} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{8}{15}x=\frac{28}{5}$$

Divide -\frac{8}{15}, the coefficient of the x term, by 2 to get -\frac{4}{15}. Then add the square of -\frac{4}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=\frac{28}{5}+\left(-\frac{4}{15}\right)^{2}$$

Square -\frac{4}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{28}{5}+\frac{16}{225}$$

Add \frac{28}{5} to \frac{16}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{1276}{225}$$

Factor x^{2}-\frac{8}{15}x+\frac{16}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{15}\right)^{2}=\frac{1276}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{\frac{1276}{225}}$$

Simplify.

$$x-\frac{4}{15}=\frac{2\sqrt{319}}{15} x-\frac{4}{15}=-\frac{2\sqrt{319}}{15}$$

Add \frac{4}{15} to both sides of the equation.

$$x=\frac{2\sqrt{319}+4}{15} x=\frac{4-2\sqrt{319}}{15}$$