Solve 7x+4/x=+4+9/5

$$7 x + \frac { 4 } { x } = + 4 + \frac { 9 } { 5 }$$

Solve for x (complex solution)

$$x=\frac{29+\sqrt{1959}i}{70}\approx 0.414285714+0.632294171i$$

$$x=\frac{-\sqrt{1959}i+29}{70}\approx 0.414285714-0.632294171i$$

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 5x, the least common multiple of x,5.

$$7x\times 5x+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

Multiply 7 and 5 to get 35.

$$35xx+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

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

$$35x^{2}+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

Multiply 5 and 4 to get 20.

$$35x^{2}+20=5x\times 4+5x\times \frac{9}{5}$$

Multiply 5 and 4 to get 20.

$$35x^{2}+20=20x+5x\times \frac{9}{5}$$

Cancel out 5 and 5.

$$35x^{2}+20=20x+9x$$

Combine 20x and 9x to get 29x.

$$35x^{2}+20=29x$$

Subtract 29x from both sides.

$$35x^{2}+20-29x=0$$

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.

$$35x^{2}-29x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, -29 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 35\times 20}}{2\times 35}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 35\times 20}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-\left(-29\right)±\sqrt{841-140\times 20}}{2\times 35}$$

Multiply -140 times 20.

$$x=\frac{-\left(-29\right)±\sqrt{841-2800}}{2\times 35}$$

Add 841 to -2800.

$$x=\frac{-\left(-29\right)±\sqrt{-1959}}{2\times 35}$$

Take the square root of -1959.

$$x=\frac{-\left(-29\right)±\sqrt{1959}i}{2\times 35}$$

The opposite of -29 is 29.

$$x=\frac{29±\sqrt{1959}i}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{29±\sqrt{1959}i}{70}$$

Now solve the equation x=\frac{29±\sqrt{1959}i}{70} when ± is plus. Add 29 to i\sqrt{1959}.

$$x=\frac{29+\sqrt{1959}i}{70}$$

Now solve the equation x=\frac{29±\sqrt{1959}i}{70} when ± is minus. Subtract i\sqrt{1959} from 29.

$$x=\frac{-\sqrt{1959}i+29}{70}$$

The equation is now solved.

$$x=\frac{29+\sqrt{1959}i}{70} x=\frac{-\sqrt{1959}i+29}{70}$$

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 5x, the least common multiple of x,5.

$$7x\times 5x+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

Multiply 7 and 5 to get 35.

$$35xx+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

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

$$35x^{2}+5\times 4=5x\times 4+5x\times \frac{9}{5}$$

Multiply 5 and 4 to get 20.

$$35x^{2}+20=5x\times 4+5x\times \frac{9}{5}$$

Multiply 5 and 4 to get 20.

$$35x^{2}+20=20x+5x\times \frac{9}{5}$$

Cancel out 5 and 5.

$$35x^{2}+20=20x+9x$$

Combine 20x and 9x to get 29x.

$$35x^{2}+20=29x$$

Subtract 29x from both sides.

$$35x^{2}+20-29x=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$35x^{2}-29x=-20$$

Divide both sides by 35.

$$\frac{35x^{2}-29x}{35}=-\frac{20}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}-\frac{29}{35}x=-\frac{20}{35}$$

Reduce the fraction \frac{-20}{35} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{29}{35}x=-\frac{4}{7}$$

Divide -\frac{29}{35}, the coefficient of the x term, by 2 to get -\frac{29}{70}. Then add the square of -\frac{29}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{35}x+\left(-\frac{29}{70}\right)^{2}=-\frac{4}{7}+\left(-\frac{29}{70}\right)^{2}$$

Square -\frac{29}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{35}x+\frac{841}{4900}=-\frac{4}{7}+\frac{841}{4900}$$

Add -\frac{4}{7} to \frac{841}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{29}{35}x+\frac{841}{4900}=-\frac{1959}{4900}$$

Factor x^{2}-\frac{29}{35}x+\frac{841}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{70}\right)^{2}=-\frac{1959}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{70}\right)^{2}}=\sqrt{-\frac{1959}{4900}}$$

Simplify.

$$x-\frac{29}{70}=\frac{\sqrt{1959}i}{70} x-\frac{29}{70}=-\frac{\sqrt{1959}i}{70}$$

Add \frac{29}{70} to both sides of the equation.

$$x=\frac{29+\sqrt{1959}i}{70} x=\frac{-\sqrt{1959}i+29}{70}$$