Solve x^2*10^-16+6.21*10^8x+1.12*10^33=0

$$x ^ { 2 } \times 10 ^ { - 16 } + 6.21 \times 10 ^ { 8 } x + 1.12 \times 10 ^ { 33 } = 0$$

Solve for x (complex solution)

$$x=-3105000000000000000000000+i\times 5000000000000000000000\sqrt{62359}\approx -3.105 \cdot 10^{24}+1.248589204 \cdot 10^{24}i$$

$$x=-i\times 5000000000000000000000\sqrt{62359}-3105000000000000000000000\approx -3.105 \cdot 10^{24}-1.248589204 \cdot 10^{24}i$$

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.

$$\frac{1}{10000000000000000}x^{2}[phone]x+1120000000000000000000000000000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{10000000000000000} for a, 621000000 for b, and 1120000000000000000000000000000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-621000000±\sqrt{621000000^{2}-4\times \frac{1}{10000000000000000}\times 1120000000000000000000000000000000}}{2\times \frac{1}{10000000000000000}}$$

Square 621000000.

$$x=\frac{-621000000±\sqrt{385641000000000000-4\times \frac{1}{10000000000000000}\times 1120000000000000000000000000000000}}{2\times \frac{1}{10000000000000000}}$$

Multiply -4 times \frac{1}{10000000000000000}.

$$x=\frac{-621000000±\sqrt{385641000000000000-\frac{1}{2500000000000000}\times 1120000000000000000000000000000000}}{2\times \frac{1}{10000000000000000}}$$

Multiply -\frac{1}{2500000000000000} times 1120000000000000000000000000000000.

$$x=\frac{-621000000±\sqrt{385641000000000000-448000000000000000}}{2\times \frac{1}{10000000000000000}}$$

Add 385641000000000000 to -448000000000000000.

$$x=\frac{-621000000±\sqrt{-62359000000000000}}{2\times \frac{1}{10000000000000000}}$$

Take the square root of -62359000000000000.

$$x=\frac{-621000000±1000000\sqrt{62359}i}{2\times \frac{1}{10000000000000000}}$$

Multiply 2 times \frac{1}{10000000000000000}.

$$x=\frac{-621000000±1000000\sqrt{62359}i}{\frac{1}{5000000000000000}}$$

Now solve the equation x=\frac{-621000000±1000000\sqrt{62359}i}{\frac{1}{5000000000000000}} when ± is plus. Add -621000000 to 1000000i\sqrt{62359}.

$$x=\frac{-621000000+1000000\sqrt{62359}i}{\frac{1}{5000000000000000}}$$

Divide -621000000+1000000i\sqrt{62359} by \frac{1}{5000000000000000} by multiplying -621000000+1000000i\sqrt{62359} by the reciprocal of \frac{1}{5000000000000000}.

$$x=-3105000000000000000000000+5000000000000000000000\sqrt{62359}i$$

Now solve the equation x=\frac{-621000000±1000000\sqrt{62359}i}{\frac{1}{5000000000000000}} when ± is minus. Subtract 1000000i\sqrt{62359} from -621000000.

$$x=\frac{-1000000\sqrt{62359}i-621000000}{\frac{1}{5000000000000000}}$$

Divide -621000000-1000000i\sqrt{62359} by \frac{1}{5000000000000000} by multiplying -621000000-1000000i\sqrt{62359} by the reciprocal of \frac{1}{5000000000000000}.

$$x=-5000000000000000000000\sqrt{62359}i-3105000000000000000000000$$

The equation is now solved.

$$x=-3105000000000000000000000+5000000000000000000000\sqrt{62359}i x=-5000000000000000000000\sqrt{62359}i-3105000000000000000000000$$

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.

$$\frac{1}{10000000000000000}x^{2}[phone]x+1120000000000000000000000000000000=0$$

Subtract 1120000000000000000000000000000000 from both sides of the equation.

$$\frac{1}{10000000000000000}x^{2}[phone]x+1120000000000000000000000000000000-1120000000000000000000000000000000=-1120000000000000000000000000000000$$

Subtracting 1120000000000000000000000000000000 from itself leaves 0.

$$\frac{1}{10000000000000000}x^{2}[phone]x=-1120000000000000000000000000000000$$

Multiply both sides by 10000000000000000.

$$\frac{\frac{1}{10000000000000000}x^{2}[phone]x}{\frac{1}{10000000000000000}}=-\frac{1120000000000000000000000000000000}{\frac{1}{10000000000000000}}$$

Dividing by \frac{1}{10000000000000000} undoes the multiplication by \frac{1}{10000000000000000}.

$$x^{2}+\frac{621000000}{\frac{1}{10000000000000000}}x=-\frac{1120000000000000000000000000000000}{\frac{1}{10000000000000000}}$$

Divide 621000000 by \frac{1}{10000000000000000} by multiplying 621000000 by the reciprocal of \frac{1}{10000000000000000}.

$$x^{2}+6210000000000000000000000x=-\frac{1120000000000000000000000000000000}{\frac{1}{10000000000000000}}$$

Divide -1120000000000000000000000000000000 by \frac{1}{10000000000000000} by multiplying -1120000000000000000000000000000000 by the reciprocal of \frac{1}{10000000000000000}.

$$x^{2}+6210000000000000000000000x=-11200000000000000000000000000000000000000000000000$$

Divide 6210000000000000000000000, the coefficient of the x term, by 2 to get 3105000000000000000000000. Then add the square of 3105000000000000000000000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6210000000000000000000000x+3105000000000000000000000^{2}=-11200000000000000000000000000000000000000000000000+3105000000000000000000000^{2}$$

Square 3105000000000000000000000.

$$x^{2}+6210000000000000000000000x+9641025000000000000000000000000000000000000000000=-11200000000000000000000000000000000000000000000000+9641025000000000000000000000000000000000000000000$$

Add -11200000000000000000000000000000000000000000000000 to 9641025000000000000000000000000000000000000000000.

$$x^{2}+6210000000000000000000000x+9641025000000000000000000000000000000000000000000=-1558975000000000000000000000000000000000000000000$$

Factor x^{2}+6210000000000000000000000x+9641025000000000000000000000000000000000000000000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3105000000000000000000000\right)^{2}=-1558975000000000000000000000000000000000000000000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3105000000000000000000000\right)^{2}}=\sqrt{-1558975000000000000000000000000000000000000000000}$$

Simplify.

$$x+3105000000000000000000000=5000000000000000000000\sqrt{62359}i x+3105000000000000000000000=-5000000000000000000000\sqrt{62359}i$$

Subtract 3105000000000000000000000 from both sides of the equation.

$$x=-3105000000000000000000000+5000000000000000000000\sqrt{62359}i x=-5000000000000000000000\sqrt{62359}i-3105000000000000000000000$$