Solve -x^2+17440^-5x+7410^-5=0

$$- x ^ { 2 } + 17440 ^ { - 5 } x + 7410 ^ { - 5 } = 0$$

Solve for x

$$x=\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}\approx 2.115701919 \cdot 10^{-10}$$

$$x=-\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}\approx -2.115701919 \cdot 10^{-10}$$

Steps Using the Quadratic Formula

Calculate 17440 to the power of -5 and get \frac{1}{1613364152133222400000}.

$$-x^{2}+\frac{1}{1613364152133222400000}x+7410^{-5}=0$$

Calculate 7410 to the power of -5 and get \frac{1}{22340404891970100000}.

$$-x^{2}+\frac{1}{1613364152133222400000}x+\frac{1}{22340404891970100000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{1}{1613364152133222400000} for b, and \frac{1}{22340404891970100000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{1613364152133222400000}±\sqrt{\left(\frac{1}{1613364152133222400000}\right)^{2}-4\left(-1\right)\times \frac{1}{22340404891970100000}}}{2\left(-1\right)}$$

Square \frac{1}{1613364152133222400000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{1613364152133222400000}±\sqrt{\frac{1}{2602943887388551592824557807861760000000000}-4\left(-1\right)\times \frac{1}{22340404891970100000}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{1}{1613364152133222400000}±\sqrt{\frac{1}{2602943887388551592824557807861760000000000}+4\times \frac{1}{22340404891970100000}}}{2\left(-1\right)}$$

Multiply 4 times \frac{1}{22340404891970100000}.

$$x=\frac{-\frac{1}{1613364152133222400000}±\sqrt{\frac{1}{2602943887388551592824557807861760000000000}+\frac{1}{5585101222992525000}}}{2\left(-1\right)}$$

Add \frac{1}{2602943887388551592824557807861760000000000} to \frac{1}{5585101222992525000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{1613364152133222400000}±\sqrt{\frac{104117755495542063712982535718519319701}{581508203553388670868992256558662048852748533760000000000}}}{2\left(-1\right)}$$

Take the square root of \frac{104117755495542063712982535718519319701}{581508203553388670868992256558662048852748533760000000000}.

$$x=\frac{-\frac{1}{1613364152133222400000}±\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{1}{1613364152133222400000}±\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}}{-2}$$

Now solve the equation x=\frac{-\frac{1}{1613364152133222400000}±\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}}{-2} when ± is plus. Add -\frac{1}{1613364152133222400000} to \frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}.

$$x=\frac{\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}-\frac{1}{1613364152133222400000}}{-2}$$

Divide -\frac{1}{1613364152133222400000}+\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000} by -2.

$$x=-\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}$$

Now solve the equation x=\frac{-\frac{1}{1613364152133222400000}±\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}}{-2} when ± is minus. Subtract \frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000} from -\frac{1}{1613364152133222400000}.

$$x=\frac{-\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000}-\frac{1}{1613364152133222400000}}{-2}$$

Divide -\frac{1}{1613364152133222400000}-\frac{\sqrt{77151256822196669211320058967422815898441}}{656427893094939259463270400000} by -2.

$$x=\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}$$

The equation is now solved.

$$x=-\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000} x=\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}$$

Steps for Completing the Square

Calculate 17440 to the power of -5 and get \frac{1}{1613364152133222400000}.

$$-x^{2}+\frac{1}{1613364152133222400000}x+7410^{-5}=0$$

Calculate 7410 to the power of -5 and get \frac{1}{22340404891970100000}.

$$-x^{2}+\frac{1}{1613364152133222400000}x+\frac{1}{22340404891970100000}=0$$

Subtract \frac{1}{22340404891970100000} from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}+\frac{1}{1613364152133222400000}x=-\frac{1}{22340404891970100000}$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{1}{1613364152133222400000}x}{-1}=-\frac{\frac{1}{22340404891970100000}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{1}{1613364152133222400000}}{-1}x=-\frac{\frac{1}{22340404891970100000}}{-1}$$

Divide \frac{1}{1613364152133222400000} by -1.

$$x^{2}-\frac{1}{1613364152133222400000}x=-\frac{\frac{1}{22340404891970100000}}{-1}$$

Divide -\frac{1}{22340404891970100000} by -1.

$$x^{2}-\frac{1}{1613364152133222400000}x=\frac{1}{22340404891970100000}$$

Divide -\frac{1}{1613364152133222400000}, the coefficient of the x term, by 2 to get -\frac{1}{3226728304266444800000}. Then add the square of -\frac{1}{3226728304266444800000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{1613364152133222400000}x+\left(-\frac{1}{3226728304266444800000}\right)^{2}=\frac{1}{22340404891970100000}+\left(-\frac{1}{3226728304266444800000}\right)^{2}$$

Square -\frac{1}{3226728304266444800000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{1613364152133222400000}x+\frac{1}{10411775549554206371298231231447040000000000}=\frac{1}{22340404891970100000}+\frac{1}{10411775549554206371298231231447040000000000}$$

Add \frac{1}{22340404891970100000} to \frac{1}{10411775549554206371298231231447040000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{1613364152133222400000}x+\frac{1}{10411775549554206371298231231447040000000000}=\frac{104117755495542063712982535718519319701}{2326032814213554683475969026234648195410994135040000000000}$$

Factor x^{2}-\frac{1}{1613364152133222400000}x+\frac{1}{10411775549554206371298231231447040000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3226728304266444800000}\right)^{2}=\frac{104117755495542063712982535718519319701}{2326032814213554683475969026234648195410994135040000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3226728304266444800000}\right)^{2}}=\sqrt{\frac{104117755495542063712982535718519319701}{2326032814213554683475969026234648195410994135040000000000}}$$

Simplify.

$$x-\frac{1}{3226728304266444800000}=\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000} x-\frac{1}{3226728304266444800000}=-\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}$$

Add \frac{1}{3226728304266444800000} to both sides of the equation.

$$x=\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000} x=-\frac{\sqrt{77151256822196669211320058967422815898441}}{1312855786189878518926540800000}+\frac{1}{3226728304266444800000}$$