Solve 7x^2+5x+3/x=14

$$\frac { 7 x ^ { 2 } + 5 x + 3 } { x } = 14$$

Solve for x (complex solution)

$$x=\frac{9+\sqrt{3}i}{14}\approx 0.642857143+0.123717915i$$

$$x=\frac{-\sqrt{3}i+9}{14}\approx 0.642857143-0.123717915i$$

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 x.

$$7x^{2}+5x+3=14x$$

Subtract 14x from both sides.

$$7x^{2}+5x+3-14x=0$$

Combine 5x and -14x to get -9x.

$$7x^{2}-9x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -9 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 7\times 3}}{2\times 7}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 7\times 3}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-9\right)±\sqrt{81-28\times 3}}{2\times 7}$$

Multiply -28 times 3.

$$x=\frac{-\left(-9\right)±\sqrt{81-84}}{2\times 7}$$

Add 81 to -84.

$$x=\frac{-\left(-9\right)±\sqrt{-3}}{2\times 7}$$

Take the square root of -3.

$$x=\frac{-\left(-9\right)±\sqrt{3}i}{2\times 7}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{3}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{9±\sqrt{3}i}{14}$$

Now solve the equation x=\frac{9±\sqrt{3}i}{14} when ± is plus. Add 9 to i\sqrt{3}.

$$x=\frac{9+\sqrt{3}i}{14}$$

Now solve the equation x=\frac{9±\sqrt{3}i}{14} when ± is minus. Subtract i\sqrt{3} from 9.

$$x=\frac{-\sqrt{3}i+9}{14}$$

The equation is now solved.

$$x=\frac{9+\sqrt{3}i}{14} x=\frac{-\sqrt{3}i+9}{14}$$

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 x.

$$7x^{2}+5x+3=14x$$

Subtract 14x from both sides.

$$7x^{2}+5x+3-14x=0$$

Combine 5x and -14x to get -9x.

$$7x^{2}-9x+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$7x^{2}-9x=-3$$

Divide both sides by 7.

$$\frac{7x^{2}-9x}{7}=-\frac{3}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{9}{7}x=-\frac{3}{7}$$

Divide -\frac{9}{7}, the coefficient of the x term, by 2 to get -\frac{9}{14}. Then add the square of -\frac{9}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{7}x+\left(-\frac{9}{14}\right)^{2}=-\frac{3}{7}+\left(-\frac{9}{14}\right)^{2}$$

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

$$x^{2}-\frac{9}{7}x+\frac{81}{196}=-\frac{3}{7}+\frac{81}{196}$$

Add -\frac{3}{7} to \frac{81}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{7}x+\frac{81}{196}=-\frac{3}{196}$$

Factor x^{2}-\frac{9}{7}x+\frac{81}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{14}\right)^{2}=-\frac{3}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{14}\right)^{2}}=\sqrt{-\frac{3}{196}}$$

Simplify.

$$x-\frac{9}{14}=\frac{\sqrt{3}i}{14} x-\frac{9}{14}=-\frac{\sqrt{3}i}{14}$$

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

$$x=\frac{9+\sqrt{3}i}{14} x=\frac{-\sqrt{3}i+9}{14}$$