Solve 2100x^2-3351x+1340=0

$$2100 { x }^{ 2 } -3351x+1340 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}\approx 0.797857143+0.038977143i$$

$$x=-\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}\approx 0.797857143-0.038977143i$$

Steps Using the Quadratic Formula

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.

$$2100x^{2}-3351x+1340=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2100 for a, -3351 for b, and 1340 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3351\right)±\sqrt{\left(-3351\right)^{2}-4\times 2100\times 1340}}{2\times 2100}$$

Square -3351.

$$x=\frac{-\left(-3351\right)±\sqrt{11229201-4\times 2100\times 1340}}{2\times 2100}$$

Multiply -4 times 2100.

$$x=\frac{-\left(-3351\right)±\sqrt{11229201-8400\times 1340}}{2\times 2100}$$

Multiply -8400 times 1340.

$$x=\frac{-\left(-3351\right)±\sqrt{11229201-11256000}}{2\times 2100}$$

Add 11229201 to -11256000.

$$x=\frac{-\left(-3351\right)±\sqrt{-26799}}{2\times 2100}$$

Take the square root of -26799.

$$x=\frac{-\left(-3351\right)±\sqrt{26799}i}{2\times 2100}$$

The opposite of -3351 is 3351.

$$x=\frac{3351±\sqrt{26799}i}{2\times 2100}$$

Multiply 2 times 2100.

$$x=\frac{3351±\sqrt{26799}i}{4200}$$

Now solve the equation x=\frac{3351±\sqrt{26799}i}{4200} when ± is plus. Add 3351 to i\sqrt{26799}.

$$x=\frac{3351+\sqrt{26799}i}{4200}$$

Divide 3351+i\sqrt{26799} by 4200.

$$x=\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}$$

Now solve the equation x=\frac{3351±\sqrt{26799}i}{4200} when ± is minus. Subtract i\sqrt{26799} from 3351.

$$x=\frac{-\sqrt{26799}i+3351}{4200}$$

Divide 3351-i\sqrt{26799} by 4200.

$$x=-\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}$$

The equation is now solved.

$$x=\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400} x=-\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}$$

Steps for Completing the Square

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.

$$2100x^{2}-3351x+1340=0$$

Subtract 1340 from both sides of the equation.

$$2100x^{2}-3351x+1340-1340=-1340$$

Subtracting 1340 from itself leaves 0.

$$2100x^{2}-3351x=-1340$$

Divide both sides by 2100.

$$\frac{2100x^{2}-3351x}{2100}=-\frac{1340}{2100}$$

Dividing by 2100 undoes the multiplication by 2100.

$$x^{2}+\left(-\frac{3351}{2100}\right)x=-\frac{1340}{2100}$$

Reduce the fraction \frac{-3351}{2100} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{1117}{700}x=-\frac{1340}{2100}$$

Reduce the fraction \frac{-1340}{2100} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{1117}{700}x=-\frac{67}{105}$$

Divide -\frac{1117}{700}, the coefficient of the x term, by 2 to get -\frac{1117}{1400}. Then add the square of -\frac{1117}{1400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1117}{700}x+\left(-\frac{1117}{1400}\right)^{2}=-\frac{67}{105}+\left(-\frac{1117}{1400}\right)^{2}$$

Square -\frac{1117}{1400} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1117}{700}x+\frac{1247689}{1960000}=-\frac{67}{105}+\frac{1247689}{1960000}$$

Add -\frac{67}{105} to \frac{1247689}{1960000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1117}{700}x+\frac{1247689}{1960000}=-\frac{8933}{5880000}$$

Factor x^{2}-\frac{1117}{700}x+\frac{1247689}{1960000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1117}{1400}\right)^{2}=-\frac{8933}{5880000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1117}{1400}\right)^{2}}=\sqrt{-\frac{8933}{5880000}}$$

Simplify.

$$x-\frac{1117}{1400}=\frac{\sqrt{26799}i}{4200} x-\frac{1117}{1400}=-\frac{\sqrt{26799}i}{4200}$$

Add \frac{1117}{1400} to both sides of the equation.

$$x=\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400} x=-\frac{\sqrt{26799}i}{4200}+\frac{1117}{1400}$$