Solve 17420x^2-7960x+2100=0

$$17420 { x }^{ 2 } -7960x+2100=0$$

Solve for x (complex solution)

$$x=\frac{199+\sqrt{51854}i}{871}\approx 0.22847302+0.261440567i$$

$$x=\frac{-\sqrt{51854}i+199}{871}\approx 0.22847302-0.261440567i$$

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.

$$17420x^{2}-7960x+2100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 17420 for a, -7960 for b, and 2100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7960\right)±\sqrt{\left(-7960\right)^{2}-4\times 17420\times 2100}}{2\times 17420}$$

Square -7960.

$$x=\frac{-\left(-7960\right)±\sqrt{63361600-4\times 17420\times 2100}}{2\times 17420}$$

Multiply -4 times 17420.

$$x=\frac{-\left(-7960\right)±\sqrt{63361600-69680\times 2100}}{2\times 17420}$$

Multiply -69680 times 2100.

$$x=\frac{-\left(-7960\right)±\sqrt{63361600-146328000}}{2\times 17420}$$

Add 63361600 to -146328000.

$$x=\frac{-\left(-7960\right)±\sqrt{-82966400}}{2\times 17420}$$

Take the square root of -82966400.

$$x=\frac{-\left(-7960\right)±40\sqrt{51854}i}{2\times 17420}$$

The opposite of -7960 is 7960.

$$x=\frac{7960±40\sqrt{51854}i}{2\times 17420}$$

Multiply 2 times 17420.

$$x=\frac{7960±40\sqrt{51854}i}{34840}$$

Now solve the equation x=\frac{7960±40\sqrt{51854}i}{34840} when ± is plus. Add 7960 to 40i\sqrt{51854}.

$$x=\frac{7960+40\sqrt{51854}i}{34840}$$

Divide 7960+40i\sqrt{51854} by 34840.

$$x=\frac{199+\sqrt{51854}i}{871}$$

Now solve the equation x=\frac{7960±40\sqrt{51854}i}{34840} when ± is minus. Subtract 40i\sqrt{51854} from 7960.

$$x=\frac{-40\sqrt{51854}i+7960}{34840}$$

Divide 7960-40i\sqrt{51854} by 34840.

$$x=\frac{-\sqrt{51854}i+199}{871}$$

The equation is now solved.

$$x=\frac{199+\sqrt{51854}i}{871} x=\frac{-\sqrt{51854}i+199}{871}$$

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.

$$17420x^{2}-7960x+2100=0$$

Subtract 2100 from both sides of the equation.

$$17420x^{2}-7960x+2100-2100=-2100$$

Subtracting 2100 from itself leaves 0.

$$17420x^{2}-7960x=-2100$$

Divide both sides by 17420.

$$\frac{17420x^{2}-7960x}{17420}=-\frac{2100}{17420}$$

Dividing by 17420 undoes the multiplication by 17420.

$$x^{2}+\left(-\frac{7960}{17420}\right)x=-\frac{2100}{17420}$$

Reduce the fraction \frac{-7960}{17420} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{398}{871}x=-\frac{2100}{17420}$$

Reduce the fraction \frac{-2100}{17420} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{398}{871}x=-\frac{105}{871}$$

Divide -\frac{398}{871}, the coefficient of the x term, by 2 to get -\frac{199}{871}. Then add the square of -\frac{199}{871} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{398}{871}x+\left(-\frac{199}{871}\right)^{2}=-\frac{105}{871}+\left(-\frac{199}{871}\right)^{2}$$

Square -\frac{199}{871} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{398}{871}x+\frac{39601}{758641}=-\frac{105}{871}+\frac{39601}{758641}$$

Add -\frac{105}{871} to \frac{39601}{758641} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{398}{871}x+\frac{39601}{758641}=-\frac{51854}{758641}$$

Factor x^{2}-\frac{398}{871}x+\frac{39601}{758641}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{199}{871}\right)^{2}=-\frac{51854}{758641}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{199}{871}\right)^{2}}=\sqrt{-\frac{51854}{758641}}$$

Simplify.

$$x-\frac{199}{871}=\frac{\sqrt{51854}i}{871} x-\frac{199}{871}=-\frac{\sqrt{51854}i}{871}$$

Add \frac{199}{871} to both sides of the equation.

$$x=\frac{199+\sqrt{51854}i}{871} x=\frac{-\sqrt{51854}i+199}{871}$$