Solve x/5+(4x/5)1/8+(x/5)frac{5}{3}+[[frac{4}{5}x]frac{1}{8}]^2+1500=x

$$\frac { x } { 5 } + ( \frac { 4 x } { 5 } ) \frac { 1 } { 8 } + ( \frac { x } { 5 } ) \frac { 5 } { 3 } + [ [ \frac { 4 } { 5 } x ] \frac { 1 } { 8 } ] ^ { 2 } + 1500 = x$$

Solve for x (complex solution)

$$x=\frac{55+5\sqrt{53879}i}{3}\approx 18.333333333+386.864173695i$$

$$x=\frac{-5\sqrt{53879}i+55}{3}\approx 18.333333333-386.864173695i$$

Steps Using the Quadratic Formula

Multiply \frac{4x}{5} times \frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x}{5}+\frac{4x}{5\times 8}+\frac{x}{5}\times \frac{5}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Cancel out 4 in both numerator and denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x}{5}\times \frac{5}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Multiply \frac{x}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x\times 5}{5\times 3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Cancel out 5 in both numerator and denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Combine \frac{x}{5} and \frac{x}{3} to get \frac{8}{15}x.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Multiply \frac{4}{5} and \frac{1}{8} to get \frac{1}{10}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{1}{10}x\right)^{2}+1500=x$$

Expand \left(\frac{1}{10}x\right)^{2}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{1}{10}\right)^{2}x^{2}+1500=x$$

Calculate \frac{1}{10} to the power of 2 and get \frac{1}{100}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\frac{1}{100}x^{2}+1500=x$$

Multiply 2 and 5 to get 10.

$$\frac{8}{15}x+\frac{x}{10}+\frac{1}{100}x^{2}+1500=x$$

Combine \frac{8}{15}x and \frac{x}{10} to get \frac{19}{30}x.

$$\frac{19}{30}x+\frac{1}{100}x^{2}+1500=x$$

Subtract x from both sides.

$$\frac{19}{30}x+\frac{1}{100}x^{2}+1500-x=0$$

Combine \frac{19}{30}x and -x to get -\frac{11}{30}x.

$$-\frac{11}{30}x+\frac{1}{100}x^{2}+1500=0$$

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.

$$\frac{1}{100}x^{2}-\frac{11}{30}x+1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{100} for a, -\frac{11}{30} for b, and 1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{11}{30}\right)±\sqrt{\left(-\frac{11}{30}\right)^{2}-4\times \frac{1}{100}\times 1500}}{2\times \frac{1}{100}}$$

Square -\frac{11}{30} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{11}{30}\right)±\sqrt{\frac{121}{900}-4\times \frac{1}{100}\times 1500}}{2\times \frac{1}{100}}$$

Multiply -4 times \frac{1}{100}.

$$x=\frac{-\left(-\frac{11}{30}\right)±\sqrt{\frac{121}{900}-\frac{1}{25}\times 1500}}{2\times \frac{1}{100}}$$

Multiply -\frac{1}{25} times 1500.

$$x=\frac{-\left(-\frac{11}{30}\right)±\sqrt{\frac{121}{900}-60}}{2\times \frac{1}{100}}$$

Add \frac{121}{900} to -60.

$$x=\frac{-\left(-\frac{11}{30}\right)±\sqrt{-\frac{53879}{900}}}{2\times \frac{1}{100}}$$

Take the square root of -\frac{53879}{900}.

$$x=\frac{-\left(-\frac{11}{30}\right)±\frac{\sqrt{53879}i}{30}}{2\times \frac{1}{100}}$$

The opposite of -\frac{11}{30} is \frac{11}{30}.

$$x=\frac{\frac{11}{30}±\frac{\sqrt{53879}i}{30}}{2\times \frac{1}{100}}$$

Multiply 2 times \frac{1}{100}.

$$x=\frac{\frac{11}{30}±\frac{\sqrt{53879}i}{30}}{\frac{1}{50}}$$

Now solve the equation x=\frac{\frac{11}{30}±\frac{\sqrt{53879}i}{30}}{\frac{1}{50}} when ± is plus. Add \frac{11}{30} to \frac{i\sqrt{53879}}{30}.

$$x=\frac{11+\sqrt{53879}i}{\frac{1}{50}\times 30}$$

Divide \frac{11+i\sqrt{53879}}{30} by \frac{1}{50} by multiplying \frac{11+i\sqrt{53879}}{30} by the reciprocal of \frac{1}{50}.

$$x=\frac{55+5\sqrt{53879}i}{3}$$

Now solve the equation x=\frac{\frac{11}{30}±\frac{\sqrt{53879}i}{30}}{\frac{1}{50}} when ± is minus. Subtract \frac{i\sqrt{53879}}{30} from \frac{11}{30}.

$$x=\frac{-\sqrt{53879}i+11}{\frac{1}{50}\times 30}$$

Divide \frac{11-i\sqrt{53879}}{30} by \frac{1}{50} by multiplying \frac{11-i\sqrt{53879}}{30} by the reciprocal of \frac{1}{50}.

$$x=\frac{-5\sqrt{53879}i+55}{3}$$

The equation is now solved.

$$x=\frac{55+5\sqrt{53879}i}{3} x=\frac{-5\sqrt{53879}i+55}{3}$$

Steps for Completing the Square

Multiply \frac{4x}{5} times \frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x}{5}+\frac{4x}{5\times 8}+\frac{x}{5}\times \frac{5}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Cancel out 4 in both numerator and denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x}{5}\times \frac{5}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Multiply \frac{x}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x\times 5}{5\times 3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Cancel out 5 in both numerator and denominator.

$$\frac{x}{5}+\frac{x}{2\times 5}+\frac{x}{3}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Combine \frac{x}{5} and \frac{x}{3} to get \frac{8}{15}x.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{4}{5}x\times \frac{1}{8}\right)^{2}+1500=x$$

Multiply \frac{4}{5} and \frac{1}{8} to get \frac{1}{10}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{1}{10}x\right)^{2}+1500=x$$

Expand \left(\frac{1}{10}x\right)^{2}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\left(\frac{1}{10}\right)^{2}x^{2}+1500=x$$

Calculate \frac{1}{10} to the power of 2 and get \frac{1}{100}.

$$\frac{8}{15}x+\frac{x}{2\times 5}+\frac{1}{100}x^{2}+1500=x$$

Multiply 2 and 5 to get 10.

$$\frac{8}{15}x+\frac{x}{10}+\frac{1}{100}x^{2}+1500=x$$

Combine \frac{8}{15}x and \frac{x}{10} to get \frac{19}{30}x.

$$\frac{19}{30}x+\frac{1}{100}x^{2}+1500=x$$

Subtract x from both sides.

$$\frac{19}{30}x+\frac{1}{100}x^{2}+1500-x=0$$

Combine \frac{19}{30}x and -x to get -\frac{11}{30}x.

$$-\frac{11}{30}x+\frac{1}{100}x^{2}+1500=0$$

Subtract 1500 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{11}{30}x+\frac{1}{100}x^{2}=-1500$$

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.

$$\frac{1}{100}x^{2}-\frac{11}{30}x=-1500$$

Multiply both sides by 100.

$$\frac{\frac{1}{100}x^{2}-\frac{11}{30}x}{\frac{1}{100}}=-\frac{1500}{\frac{1}{100}}$$

Dividing by \frac{1}{100} undoes the multiplication by \frac{1}{100}.

$$x^{2}+\left(-\frac{\frac{11}{30}}{\frac{1}{100}}\right)x=-\frac{1500}{\frac{1}{100}}$$

Divide -\frac{11}{30} by \frac{1}{100} by multiplying -\frac{11}{30} by the reciprocal of \frac{1}{100}.

$$x^{2}-\frac{110}{3}x=-\frac{1500}{\frac{1}{100}}$$

Divide -1500 by \frac{1}{100} by multiplying -1500 by the reciprocal of \frac{1}{100}.

$$x^{2}-\frac{110}{3}x=-150000$$

Divide -\frac{110}{3}, the coefficient of the x term, by 2 to get -\frac{55}{3}. Then add the square of -\frac{55}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=-150000+\left(-\frac{55}{3}\right)^{2}$$

Square -\frac{55}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=-150000+\frac{3025}{9}$$

Add -150000 to \frac{3025}{9}.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=-\frac{1346975}{9}$$

Factor x^{2}-\frac{110}{3}x+\frac{3025}{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}.

$$\left(x-\frac{55}{3}\right)^{2}=-\frac{1346975}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{-\frac{1346975}{9}}$$

Simplify.

$$x-\frac{55}{3}=\frac{5\sqrt{53879}i}{3} x-\frac{55}{3}=-\frac{5\sqrt{53879}i}{3}$$

Add \frac{55}{3} to both sides of the equation.

$$x=\frac{55+5\sqrt{53879}i}{3} x=\frac{-5\sqrt{53879}i+55}{3}$$