Solve x^2+3.394*x+3.976=0

$${ x }^{ 2 } +3.394 \times x+3.976=0$$

Solve for x (complex solution)

$$x=\frac{-1697+i\times 3\sqrt{121799}}{1000}\approx -1.697+1.046991404i$$

$$x=\frac{-i\times 3\sqrt{121799}-1697}{1000}\approx -1.697-1.046991404i$$

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.

$$x^{2}+3.394x+3.976=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3.394 for b, and 3.976 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3.394±\sqrt{3.394^{2}-4\times 3.976}}{2}$$

Square 3.394 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3.394±\sqrt{11.519236-4\times 3.976}}{2}$$

Multiply -4 times 3.976.

$$x=\frac{-3.394±\sqrt{11.519236-15.904}}{2}$$

Add 11.519236 to -15.904 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.394±\sqrt{-4.384764}}{2}$$

Take the square root of -4.384764.

$$x=\frac{-3.394±\frac{3\sqrt{121799}i}{500}}{2}$$

Now solve the equation x=\frac{-3.394±\frac{3\sqrt{121799}i}{500}}{2} when ± is plus. Add -3.394 to \frac{3i\sqrt{121799}}{500}.

$$x=\frac{-1697+3\sqrt{121799}i}{2\times 500}$$

Divide \frac{-1697+3i\sqrt{121799}}{500} by 2.

$$x=\frac{-1697+3\sqrt{121799}i}{1000}$$

Now solve the equation x=\frac{-3.394±\frac{3\sqrt{121799}i}{500}}{2} when ± is minus. Subtract \frac{3i\sqrt{121799}}{500} from -3.394.

$$x=\frac{-3\sqrt{121799}i-1697}{2\times 500}$$

Divide \frac{-1697-3i\sqrt{121799}}{500} by 2.

$$x=\frac{-3\sqrt{121799}i-1697}{1000}$$

The equation is now solved.

$$x=\frac{-1697+3\sqrt{121799}i}{1000} x=\frac{-3\sqrt{121799}i-1697}{1000}$$

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.

$$x^{2}+3.394x+3.976=0$$

Subtract 3.976 from both sides of the equation.

$$x^{2}+3.394x+3.976-3.976=-3.976$$

Subtracting 3.976 from itself leaves 0.

$$x^{2}+3.394x=-3.976$$

Divide 3.394, the coefficient of the x term, by 2 to get 1.697. Then add the square of 1.697 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3.394x+1.697^{2}=-3.976+1.697^{2}$$

Square 1.697 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3.394x+2.879809=-3.976+2.879809$$

Add -3.976 to 2.879809 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3.394x+2.879809=-1.096191$$

Factor x^{2}+3.394x+2.879809. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1.697\right)^{2}=-1.096191$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1.697\right)^{2}}=\sqrt{-1.096191}$$

Simplify.

$$x+1.697=\frac{3\sqrt{121799}i}{1000} x+1.697=-\frac{3\sqrt{121799}i}{1000}$$

Subtract 1.697 from both sides of the equation.

$$x=\frac{-1697+3\sqrt{121799}i}{1000} x=\frac{-3\sqrt{121799}i-1697}{1000}$$