Solve x^2+3394*x+3976=0

$${ x }^{ 2 } +3394 \times x+3976=0$$

Solve for x

$$x=3\sqrt{319537}-1697\approx -1.17188371$$

$$x=-3\sqrt{319537}-1697\approx -3392.82811629$$

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.

$$x^{2}+3394x+3976=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3394 for b, and 3976 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3394±\sqrt{3394^{2}-4\times 3976}}{2}$$

Square 3394.

$$x=\frac{-3394±\sqrt{11519236-4\times 3976}}{2}$$

Multiply -4 times 3976.

$$x=\frac{-3394±\sqrt{11519236-15904}}{2}$$

Add 11519236 to -15904.

$$x=\frac{-3394±\sqrt{11503332}}{2}$$

Take the square root of 11503332.

$$x=\frac{-3394±6\sqrt{319537}}{2}$$

Now solve the equation x=\frac{-3394±6\sqrt{319537}}{2} when ± is plus. Add -3394 to 6\sqrt{319537}.

$$x=\frac{6\sqrt{319537}-3394}{2}$$

Divide -3394+6\sqrt{319537} by 2.

$$x=3\sqrt{319537}-1697$$

Now solve the equation x=\frac{-3394±6\sqrt{319537}}{2} when ± is minus. Subtract 6\sqrt{319537} from -3394.

$$x=\frac{-6\sqrt{319537}-3394}{2}$$

Divide -3394-6\sqrt{319537} by 2.

$$x=-3\sqrt{319537}-1697$$

The equation is now solved.

$$x=3\sqrt{319537}-1697 x=-3\sqrt{319537}-1697$$

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.

$$x^{2}+3394x+3976=0$$

Subtract 3976 from both sides of the equation.

$$x^{2}+3394x+3976-3976=-3976$$

Subtracting 3976 from itself leaves 0.

$$x^{2}+3394x=-3976$$

Divide 3394, the coefficient of the x term, by 2 to get 1697. Then add the square of 1697 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3394x+1697^{2}=-3976+1697^{2}$$

Square 1697.

$$x^{2}+3394x+2879809=-3976+2879809$$

Add -3976 to 2879809.

$$x^{2}+3394x+2879809=2875833$$

Factor x^{2}+3394x+2879809. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1697\right)^{2}=2875833$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1697\right)^{2}}=\sqrt{2875833}$$

Simplify.

$$x+1697=3\sqrt{319537} x+1697=-3\sqrt{319537}$$

Subtract 1697 from both sides of the equation.

$$x=3\sqrt{319537}-1697 x=-3\sqrt{319537}-1697$$