Solve 1084^2+108x^2=1

$$1084 ^ { 2 } + 108 x ^ { 2 } = 1$$

Solve for x (complex solution)

$$x=-\frac{19\sqrt{1085}i}{6}\approx -0-104.307904249i$$

$$x=\frac{19\sqrt{1085}i}{6}\approx 104.307904249i$$

Steps by Finding Square Root

Calculate 1084 to the power of 2 and get 1175056.

$$1175056+108x^{2}=1$$

Subtract 1175056 from both sides.

$$108x^{2}=1-1175056$$

Subtract 1175056 from 1 to get -1175055.

$$108x^{2}=-1175055$$

Divide both sides by 108.

$$x^{2}=\frac{-1175055}{108}$$

Reduce the fraction \frac{-1175055}{108} to lowest terms by extracting and canceling out 3.

$$x^{2}=-\frac{391685}{36}$$

The equation is now solved.

$$x=\frac{19\sqrt{1085}i}{6} x=-\frac{19\sqrt{1085}i}{6}$$

Steps Using the Quadratic Formula

Calculate 1084 to the power of 2 and get 1175056.

$$1175056+108x^{2}=1$$

Subtract 1 from both sides.

$$1175056+108x^{2}-1=0$$

Subtract 1 from 1175056 to get 1175055.

$$1175055+108x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$108x^{2}+1175055=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 108 for a, 0 for b, and 1175055 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 108\times 1175055}}{2\times 108}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 108\times 1175055}}{2\times 108}$$

Multiply -4 times 108.

$$x=\frac{0±\sqrt{-432\times 1175055}}{2\times 108}$$

Multiply -432 times 1175055.

$$x=\frac{0±\sqrt{-507623760}}{2\times 108}$$

Take the square root of -507623760.

$$x=\frac{0±684\sqrt{1085}i}{2\times 108}$$

Multiply 2 times 108.

$$x=\frac{0±684\sqrt{1085}i}{216}$$

Now solve the equation x=\frac{0±684\sqrt{1085}i}{216} when ± is plus.

$$x=\frac{19\sqrt{1085}i}{6}$$

Now solve the equation x=\frac{0±684\sqrt{1085}i}{216} when ± is minus.

$$x=-\frac{19\sqrt{1085}i}{6}$$

The equation is now solved.

$$x=\frac{19\sqrt{1085}i}{6} x=-\frac{19\sqrt{1085}i}{6}$$