Solve 4=8/9x+14x

$$4= \frac{ 8 }{ 9x } +14x$$

Solve for x (complex solution)

$$x=\frac{\sqrt{19}i}{21}+\frac{1}{7}\approx 0.142857143+0.207566616i$$

$$x=-\frac{\sqrt{19}i}{21}+\frac{1}{7}\approx 0.142857143-0.207566616i$$

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 9x.

$$36x=8+14x\times 9x$$

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

$$36x=8+14x^{2}\times 9$$

Multiply 14 and 9 to get 126.

$$36x=8+126x^{2}$$

Subtract 8 from both sides.

$$36x-8=126x^{2}$$

Subtract 126x^{2} from both sides.

$$36x-8-126x^{2}=0$$

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.

$$-126x^{2}+36x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -126 for a, 36 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\left(-126\right)\left(-8\right)}}{2\left(-126\right)}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\left(-126\right)\left(-8\right)}}{2\left(-126\right)}$$

Multiply -4 times -126.

$$x=\frac{-36±\sqrt{1296+504\left(-8\right)}}{2\left(-126\right)}$$

Multiply 504 times -8.

$$x=\frac{-36±\sqrt{1296-4032}}{2\left(-126\right)}$$

Add 1296 to -4032.

$$x=\frac{-36±\sqrt{-2736}}{2\left(-126\right)}$$

Take the square root of -2736.

$$x=\frac{-36±12\sqrt{19}i}{2\left(-126\right)}$$

Multiply 2 times -126.

$$x=\frac{-36±12\sqrt{19}i}{-252}$$

Now solve the equation x=\frac{-36±12\sqrt{19}i}{-252} when ± is plus. Add -36 to 12i\sqrt{19}.

$$x=\frac{-36+12\sqrt{19}i}{-252}$$

Divide -36+12i\sqrt{19} by -252.

$$x=-\frac{\sqrt{19}i}{21}+\frac{1}{7}$$

Now solve the equation x=\frac{-36±12\sqrt{19}i}{-252} when ± is minus. Subtract 12i\sqrt{19} from -36.

$$x=\frac{-12\sqrt{19}i-36}{-252}$$

Divide -36-12i\sqrt{19} by -252.

$$x=\frac{\sqrt{19}i}{21}+\frac{1}{7}$$

The equation is now solved.

$$x=-\frac{\sqrt{19}i}{21}+\frac{1}{7} x=\frac{\sqrt{19}i}{21}+\frac{1}{7}$$

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 9x.

$$36x=8+14x\times 9x$$

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

$$36x=8+14x^{2}\times 9$$

Multiply 14 and 9 to get 126.

$$36x=8+126x^{2}$$

Subtract 126x^{2} from both sides.

$$36x-126x^{2}=8$$

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.

$$-126x^{2}+36x=8$$

Divide both sides by -126.

$$\frac{-126x^{2}+36x}{-126}=\frac{8}{-126}$$

Dividing by -126 undoes the multiplication by -126.

$$x^{2}+\frac{36}{-126}x=\frac{8}{-126}$$

Reduce the fraction \frac{36}{-126} to lowest terms by extracting and canceling out 18.

$$x^{2}-\frac{2}{7}x=\frac{8}{-126}$$

Reduce the fraction \frac{8}{-126} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{2}{7}x=-\frac{4}{63}$$

Divide -\frac{2}{7}, the coefficient of the x term, by 2 to get -\frac{1}{7}. Then add the square of -\frac{1}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=-\frac{4}{63}+\left(-\frac{1}{7}\right)^{2}$$

Square -\frac{1}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=-\frac{4}{63}+\frac{1}{49}$$

Add -\frac{4}{63} to \frac{1}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=-\frac{19}{441}$$

Factor x^{2}-\frac{2}{7}x+\frac{1}{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}.

$$\left(x-\frac{1}{7}\right)^{2}=-\frac{19}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{-\frac{19}{441}}$$

Simplify.

$$x-\frac{1}{7}=\frac{\sqrt{19}i}{21} x-\frac{1}{7}=-\frac{\sqrt{19}i}{21}$$

Add \frac{1}{7} to both sides of the equation.

$$x=\frac{\sqrt{19}i}{21}+\frac{1}{7} x=-\frac{\sqrt{19}i}{21}+\frac{1}{7}$$