Solve for t
t = \frac{\sqrt{22441} + 229}{5} \approx 75.760640848
t = \frac{229 - \sqrt{22441}}{5} \approx 15.839359152
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-500t^{2}+45800t=600000
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.
-500t^{2}+45800t-600000=600000-600000
Subtract 600000 from both sides of the equation.
-500t^{2}+45800t-600000=0
Subtracting 600000 from itself leaves 0.
t=\frac{-45800±\sqrt{45800^{2}-4\left(-500\right)\left(-600000\right)}}{2\left(-500\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -500 for a, 45800 for b, and -600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-45800±\sqrt{2097640000-4\left(-500\right)\left(-600000\right)}}{2\left(-500\right)}
Square 45800.
t=\frac{-45800±\sqrt{2097640000+2000\left(-600000\right)}}{2\left(-500\right)}
Multiply -4 times -500.
t=\frac{-45800±\sqrt{2097640000-1200000000}}{2\left(-500\right)}
Multiply 2000 times -600000.
t=\frac{-45800±\sqrt{897640000}}{2\left(-500\right)}
Add 2097640000 to -1200000000.
t=\frac{-45800±200\sqrt{22441}}{2\left(-500\right)}
Take the square root of 897640000.
t=\frac{-45800±200\sqrt{22441}}{-1000}
Multiply 2 times -500.
t=\frac{200\sqrt{22441}-45800}{-1000}
Now solve the equation t=\frac{-45800±200\sqrt{22441}}{-1000} when ± is plus. Add -45800 to 200\sqrt{22441}.
t=\frac{229-\sqrt{22441}}{5}
Divide -45800+200\sqrt{22441} by -1000.
t=\frac{-200\sqrt{22441}-45800}{-1000}
Now solve the equation t=\frac{-45800±200\sqrt{22441}}{-1000} when ± is minus. Subtract 200\sqrt{22441} from -45800.
t=\frac{\sqrt{22441}+229}{5}
Divide -45800-200\sqrt{22441} by -1000.
t=\frac{229-\sqrt{22441}}{5} t=\frac{\sqrt{22441}+229}{5}
The equation is now solved.
-500t^{2}+45800t=600000
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{-500t^{2}+45800t}{-500}=\frac{600000}{-500}
Divide both sides by -500.
t^{2}+\frac{45800}{-500}t=\frac{600000}{-500}
Dividing by -500 undoes the multiplication by -500.
t^{2}-\frac{458}{5}t=\frac{600000}{-500}
Reduce the fraction \frac{45800}{-500} to lowest terms by extracting and canceling out 100.
t^{2}-\frac{458}{5}t=-1200
Divide 600000 by -500.
t^{2}-\frac{458}{5}t+\left(-\frac{229}{5}\right)^{2}=-1200+\left(-\frac{229}{5}\right)^{2}
Divide -\frac{458}{5}, the coefficient of the x term, by 2 to get -\frac{229}{5}. Then add the square of -\frac{229}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-\frac{458}{5}t+\frac{52441}{25}=-1200+\frac{52441}{25}
Square -\frac{229}{5} by squaring both the numerator and the denominator of the fraction.
t^{2}-\frac{458}{5}t+\frac{52441}{25}=\frac{22441}{25}
Add -1200 to \frac{52441}{25}.
\left(t-\frac{229}{5}\right)^{2}=\frac{22441}{25}
Factor t^{2}-\frac{458}{5}t+\frac{52441}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(t-\frac{229}{5}\right)^{2}}=\sqrt{\frac{22441}{25}}
Take the square root of both sides of the equation.
t-\frac{229}{5}=\frac{\sqrt{22441}}{5} t-\frac{229}{5}=-\frac{\sqrt{22441}}{5}
Simplify.
t=\frac{\sqrt{22441}+229}{5} t=\frac{229-\sqrt{22441}}{5}
Add \frac{229}{5} to both sides of the equation.
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Limits
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