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-4.9t^{2}+97t+234=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{-97±\sqrt{97^{2}-4\left(-4.9\right)\times 234}}{2\left(-4.9\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 97 for b, and 234 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-97±\sqrt{9409-4\left(-4.9\right)\times 234}}{2\left(-4.9\right)}
Square 97.
t=\frac{-97±\sqrt{9409+19.6\times 234}}{2\left(-4.9\right)}
Multiply -4 times -4.9.
t=\frac{-97±\sqrt{9409+4586.4}}{2\left(-4.9\right)}
Multiply 19.6 times 234.
t=\frac{-97±\sqrt{13995.4}}{2\left(-4.9\right)}
Add 9409 to 4586.4.
t=\frac{-97±\frac{\sqrt{349885}}{5}}{2\left(-4.9\right)}
Take the square root of 13995.4.
t=\frac{-97±\frac{\sqrt{349885}}{5}}{-9.8}
Multiply 2 times -4.9.
t=\frac{\frac{\sqrt{349885}}{5}-97}{-9.8}
Now solve the equation t=\frac{-97±\frac{\sqrt{349885}}{5}}{-9.8} when ± is plus. Add -97 to \frac{\sqrt{349885}}{5}.
t=\frac{485-\sqrt{349885}}{49}
Divide -97+\frac{\sqrt{349885}}{5} by -9.8 by multiplying -97+\frac{\sqrt{349885}}{5} by the reciprocal of -9.8.
t=\frac{-\frac{\sqrt{349885}}{5}-97}{-9.8}
Now solve the equation t=\frac{-97±\frac{\sqrt{349885}}{5}}{-9.8} when ± is minus. Subtract \frac{\sqrt{349885}}{5} from -97.
t=\frac{\sqrt{349885}+485}{49}
Divide -97-\frac{\sqrt{349885}}{5} by -9.8 by multiplying -97-\frac{\sqrt{349885}}{5} by the reciprocal of -9.8.
t=\frac{485-\sqrt{349885}}{49} t=\frac{\sqrt{349885}+485}{49}
The equation is now solved.
-4.9t^{2}+97t+234=0
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.
-4.9t^{2}+97t+234-234=-234
Subtract 234 from both sides of the equation.
-4.9t^{2}+97t=-234
Subtracting 234 from itself leaves 0.
\frac{-4.9t^{2}+97t}{-4.9}=-\frac{234}{-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{97}{-4.9}t=-\frac{234}{-4.9}
Dividing by -4.9 undoes the multiplication by -4.9.
t^{2}-\frac{970}{49}t=-\frac{234}{-4.9}
Divide 97 by -4.9 by multiplying 97 by the reciprocal of -4.9.
t^{2}-\frac{970}{49}t=\frac{2340}{49}
Divide -234 by -4.9 by multiplying -234 by the reciprocal of -4.9.
t^{2}-\frac{970}{49}t+\left(-\frac{485}{49}\right)^{2}=\frac{2340}{49}+\left(-\frac{485}{49}\right)^{2}
Divide -\frac{970}{49}, the coefficient of the x term, by 2 to get -\frac{485}{49}. Then add the square of -\frac{485}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-\frac{970}{49}t+\frac{235225}{2401}=\frac{2340}{49}+\frac{235225}{2401}
Square -\frac{485}{49} by squaring both the numerator and the denominator of the fraction.
t^{2}-\frac{970}{49}t+\frac{235225}{2401}=\frac{349885}{2401}
Add \frac{2340}{49} to \frac{235225}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t-\frac{485}{49}\right)^{2}=\frac{349885}{2401}
Factor t^{2}-\frac{970}{49}t+\frac{235225}{2401}. 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{485}{49}\right)^{2}}=\sqrt{\frac{349885}{2401}}
Take the square root of both sides of the equation.
t-\frac{485}{49}=\frac{\sqrt{349885}}{49} t-\frac{485}{49}=-\frac{\sqrt{349885}}{49}
Simplify.
t=\frac{\sqrt{349885}+485}{49} t=\frac{485-\sqrt{349885}}{49}
Add \frac{485}{49} to both sides of the equation.