Solve 172700x=1254800x^2-1/628Rightarrow

$$172700 x = 1254800 x ^ { 2 } - \frac { 1 } { 628 } \Rightarrow$$

Solve for x

$$x=\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}\approx 0.137631504$$

$$x=-\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}\approx -9.220362376 \cdot 10^{-9}$$

Steps Using the Quadratic Formula

Subtract 1254800x^{2} from both sides.

$$172700x-1254800x^{2}=-\frac{1}{628}$$

Add \frac{1}{628} to both sides.

$$172700x-1254800x^{2}+\frac{1}{628}=0$$

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.

$$-1254800x^{2}+172700x+\frac{1}{628}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1254800 for a, 172700 for b, and \frac{1}{628} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-172700±\sqrt{172700^{2}-4\left(-1254800\right)\times \frac{1}{628}}}{2\left(-1254800\right)}$$

Square 172700.

$$x=\frac{-172700±\sqrt{29825290000-4\left(-1254800\right)\times \frac{1}{628}}}{2\left(-1254800\right)}$$

Multiply -4 times -1254800.

$$x=\frac{-172700±\sqrt{29825290000+5019200\times \frac{1}{628}}}{2\left(-1254800\right)}$$

Multiply 5019200 times \frac{1}{628}.

$$x=\frac{-172700±\sqrt{29825290000+\frac{1254800}{157}}}{2\left(-1254800\right)}$$

Add 29825290000 to \frac{1254800}{157}.

$$x=\frac{-172700±\sqrt{\frac{4682571784800}{157}}}{2\left(-1254800\right)}$$

Take the square root of \frac{4682571784800}{157}.

$$x=\frac{-172700±\frac{20\sqrt{1837909425534}}{157}}{2\left(-1254800\right)}$$

Multiply 2 times -1254800.

$$x=\frac{-172700±\frac{20\sqrt{1837909425534}}{157}}{-2509600}$$

Now solve the equation x=\frac{-172700±\frac{20\sqrt{1837909425534}}{157}}{-2509600} when ± is plus. Add -172700 to \frac{20\sqrt{1837909425534}}{157}.

$$x=\frac{\frac{20\sqrt{1837909425534}}{157}-172700}{-2509600}$$

Divide -172700+\frac{20\sqrt{1837909425534}}{157} by -2509600.

$$x=-\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}$$

Now solve the equation x=\frac{-172700±\frac{20\sqrt{1837909425534}}{157}}{-2509600} when ± is minus. Subtract \frac{20\sqrt{1837909425534}}{157} from -172700.

$$x=\frac{-\frac{20\sqrt{1837909425534}}{157}-172700}{-2509600}$$

Divide -172700-\frac{20\sqrt{1837909425534}}{157} by -2509600.

$$x=\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}$$

The equation is now solved.

$$x=-\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096} x=\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}$$

Steps for Completing the Square

Subtract 1254800x^{2} from both sides.

$$172700x-1254800x^{2}=-\frac{1}{628}$$

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.

$$-1254800x^{2}+172700x=-\frac{1}{628}$$

Divide both sides by -1254800.

$$\frac{-1254800x^{2}+172700x}{-1254800}=-\frac{\frac{1}{628}}{-1254800}$$

Dividing by -1254800 undoes the multiplication by -1254800.

$$x^{2}+\frac{172700}{-1254800}x=-\frac{\frac{1}{628}}{-1254800}$$

Reduce the fraction \frac{172700}{-1254800} to lowest terms by extracting and canceling out 100.

$$x^{2}-\frac{1727}{12548}x=-\frac{\frac{1}{628}}{-1254800}$$

Divide -\frac{1}{628} by -1254800.

$$x^{2}-\frac{1727}{12548}x=\frac{1}{788014400}$$

Divide -\frac{1727}{12548}, the coefficient of the x term, by 2 to get -\frac{1727}{25096}. Then add the square of -\frac{1727}{25096} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1727}{12548}x+\left(-\frac{1727}{25096}\right)^{2}=\frac{1}{788014400}+\left(-\frac{1727}{25096}\right)^{2}$$

Square -\frac{1727}{25096} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1727}{12548}x+\frac{2982529}{629809216}=\frac{1}{788014400}+\frac{2982529}{629809216}$$

Add \frac{1}{788014400} to \frac{2982529}{629809216} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1727}{12548}x+\frac{2982529}{629809216}=\frac{5853214731}{1236000586400}$$

Factor x^{2}-\frac{1727}{12548}x+\frac{2982529}{629809216}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1727}{25096}\right)^{2}=\frac{5853214731}{1236000586400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1727}{25096}\right)^{2}}=\sqrt{\frac{5853214731}{1236000586400}}$$

Simplify.

$$x-\frac{1727}{25096}=\frac{\sqrt{1837909425534}}{19700360} x-\frac{1727}{25096}=-\frac{\sqrt{1837909425534}}{19700360}$$

Add \frac{1727}{25096} to both sides of the equation.

$$x=\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096} x=-\frac{\sqrt{1837909425534}}{19700360}+\frac{1727}{25096}$$