Solve 1/7x^2+1=3/7x

$$\frac { 1 } { 7 } x ^ { 2 } + 1 = \frac { 3 } { 7 } x$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{19}i}{2}\approx 1.5+2.179449472i$$

$$x=\frac{-\sqrt{19}i+3}{2}\approx 1.5-2.179449472i$$

Steps Using the Quadratic Formula

Subtract \frac{3}{7}x from both sides.

$$\frac{1}{7}x^{2}+1-\frac{3}{7}x=0$$

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.

$$\frac{1}{7}x^{2}-\frac{3}{7}x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{7} for a, -\frac{3}{7} for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{7}\right)±\sqrt{\left(-\frac{3}{7}\right)^{2}-4\times \frac{1}{7}}}{2\times \frac{1}{7}}$$

Square -\frac{3}{7} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{3}{7}\right)±\sqrt{\frac{9}{49}-4\times \frac{1}{7}}}{2\times \frac{1}{7}}$$

Multiply -4 times \frac{1}{7}.

$$x=\frac{-\left(-\frac{3}{7}\right)±\sqrt{\frac{9}{49}-\frac{4}{7}}}{2\times \frac{1}{7}}$$

Add \frac{9}{49} to -\frac{4}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{3}{7}\right)±\sqrt{-\frac{19}{49}}}{2\times \frac{1}{7}}$$

Take the square root of -\frac{19}{49}.

$$x=\frac{-\left(-\frac{3}{7}\right)±\frac{\sqrt{19}i}{7}}{2\times \frac{1}{7}}$$

The opposite of -\frac{3}{7} is \frac{3}{7}.

$$x=\frac{\frac{3}{7}±\frac{\sqrt{19}i}{7}}{2\times \frac{1}{7}}$$

Multiply 2 times \frac{1}{7}.

$$x=\frac{\frac{3}{7}±\frac{\sqrt{19}i}{7}}{\frac{2}{7}}$$

Now solve the equation x=\frac{\frac{3}{7}±\frac{\sqrt{19}i}{7}}{\frac{2}{7}} when ± is plus. Add \frac{3}{7} to \frac{i\sqrt{19}}{7}.

$$x=\frac{3+\sqrt{19}i}{\frac{2}{7}\times 7}$$

Divide \frac{3+i\sqrt{19}}{7} by \frac{2}{7} by multiplying \frac{3+i\sqrt{19}}{7} by the reciprocal of \frac{2}{7}.

$$x=\frac{3+\sqrt{19}i}{2}$$

Now solve the equation x=\frac{\frac{3}{7}±\frac{\sqrt{19}i}{7}}{\frac{2}{7}} when ± is minus. Subtract \frac{i\sqrt{19}}{7} from \frac{3}{7}.

$$x=\frac{-\sqrt{19}i+3}{\frac{2}{7}\times 7}$$

Divide \frac{3-i\sqrt{19}}{7} by \frac{2}{7} by multiplying \frac{3-i\sqrt{19}}{7} by the reciprocal of \frac{2}{7}.

$$x=\frac{-\sqrt{19}i+3}{2}$$

The equation is now solved.

$$x=\frac{3+\sqrt{19}i}{2} x=\frac{-\sqrt{19}i+3}{2}$$

Steps for Completing the Square

Subtract \frac{3}{7}x from both sides.

$$\frac{1}{7}x^{2}+1-\frac{3}{7}x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{7}x^{2}-\frac{3}{7}x=-1$$

Multiply both sides by 7.

$$\frac{\frac{1}{7}x^{2}-\frac{3}{7}x}{\frac{1}{7}}=-\frac{1}{\frac{1}{7}}$$

Dividing by \frac{1}{7} undoes the multiplication by \frac{1}{7}.

$$x^{2}+\left(-\frac{\frac{3}{7}}{\frac{1}{7}}\right)x=-\frac{1}{\frac{1}{7}}$$

Divide -\frac{3}{7} by \frac{1}{7} by multiplying -\frac{3}{7} by the reciprocal of \frac{1}{7}.

$$x^{2}-3x=-\frac{1}{\frac{1}{7}}$$

Divide -1 by \frac{1}{7} by multiplying -1 by the reciprocal of \frac{1}{7}.

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

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-7+\frac{9}{4}$$

Add -7 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{19}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{19}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{19}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{19}i}{2} x-\frac{3}{2}=-\frac{\sqrt{19}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{3+\sqrt{19}i}{2} x=\frac{-\sqrt{19}i+3}{2}$$