- 36,34 = 11,11 t - 4,9 t ^ { 2 }
Solve for t
t = \frac{\sqrt{8356961} + 1111}{980} \approx 4.083511103
t=\frac{1111-\sqrt{8356961}}{980}\approx -1.816164164
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11,11t-4,9t^{2}=-36,34
Swap sides so that all variable terms are on the left hand side.
11,11t-4,9t^{2}+36,34=0
Add 36,34 to both sides.
-4,9t^{2}+11,11t+36,34=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.
t=\frac{-11,11±\sqrt{11,11^{2}-4\left(-4,9\right)\times 36,34}}{2\left(-4,9\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4,9 for a, 11,11 for b, and 36,34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-11,11±\sqrt{123,4321-4\left(-4,9\right)\times 36,34}}{2\left(-4,9\right)}
Square 11,11 by squaring both the numerator and the denominator of the fraction.
t=\frac{-11,11±\sqrt{123,4321+19,6\times 36,34}}{2\left(-4,9\right)}
Multiply -4 times -4,9.
t=\frac{-11,11±\sqrt{123,4321+712,264}}{2\left(-4,9\right)}
Multiply 19,6 times 36,34 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
t=\frac{-11,11±\sqrt{835,6961}}{2\left(-4,9\right)}
Add 123,4321 to 712,264 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
t=\frac{-11,11±\frac{\sqrt{8356961}}{100}}{2\left(-4,9\right)}
Take the square root of 835,6961.
t=\frac{-11,11±\frac{\sqrt{8356961}}{100}}{-9,8}
Multiply 2 times -4,9.
t=\frac{\sqrt{8356961}-1111}{-9,8\times 100}
Now solve the equation t=\frac{-11,11±\frac{\sqrt{8356961}}{100}}{-9,8} when ± is plus. Add -11,11 to \frac{\sqrt{8356961}}{100}.
t=\frac{1111-\sqrt{8356961}}{980}
Divide \frac{-1111+\sqrt{8356961}}{100} by -9,8 by multiplying \frac{-1111+\sqrt{8356961}}{100} by the reciprocal of -9,8.
t=\frac{-\sqrt{8356961}-1111}{-9,8\times 100}
Now solve the equation t=\frac{-11,11±\frac{\sqrt{8356961}}{100}}{-9,8} when ± is minus. Subtract \frac{\sqrt{8356961}}{100} from -11,11.
t=\frac{\sqrt{8356961}+1111}{980}
Divide \frac{-1111-\sqrt{8356961}}{100} by -9,8 by multiplying \frac{-1111-\sqrt{8356961}}{100} by the reciprocal of -9,8.
t=\frac{1111-\sqrt{8356961}}{980} t=\frac{\sqrt{8356961}+1111}{980}
The equation is now solved.
11,11t-4,9t^{2}=-36,34
Swap sides so that all variable terms are on the left hand side.
-4,9t^{2}+11,11t=-36,34
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{-4,9t^{2}+11,11t}{-4,9}=-\frac{36,34}{-4,9}
Divide both sides of the equation by -4,9, which is the same as multiplying both sides by the reciprocal of the fraction.
t^{2}+\frac{11,11}{-4,9}t=-\frac{36,34}{-4,9}
Dividing by -4,9 undoes the multiplication by -4,9.
t^{2}-\frac{1111}{490}t=-\frac{36,34}{-4,9}
Divide 11,11 by -4,9 by multiplying 11,11 by the reciprocal of -4,9.
t^{2}-\frac{1111}{490}t=\frac{1817}{245}
Divide -36,34 by -4,9 by multiplying -36,34 by the reciprocal of -4,9.
t^{2}-\frac{1111}{490}t+\left(-\frac{1111}{980}\right)^{2}=\frac{1817}{245}+\left(-\frac{1111}{980}\right)^{2}
Divide -\frac{1111}{490}, the coefficient of the x term, by 2 to get -\frac{1111}{980}. Then add the square of -\frac{1111}{980} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{1817}{245}+\frac{1234321}{960400}
Square -\frac{1111}{980} by squaring both the numerator and the denominator of the fraction.
t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{8356961}{960400}
Add \frac{1817}{245} to \frac{1234321}{960400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t-\frac{1111}{980}\right)^{2}=\frac{8356961}{960400}
Factor t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}. 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{1111}{980}\right)^{2}}=\sqrt{\frac{8356961}{960400}}
Take the square root of both sides of the equation.
t-\frac{1111}{980}=\frac{\sqrt{8356961}}{980} t-\frac{1111}{980}=-\frac{\sqrt{8356961}}{980}
Simplify.
t=\frac{\sqrt{8356961}+1111}{980} t=\frac{1111-\sqrt{8356961}}{980}
Add \frac{1111}{980} to both sides of the equation.
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