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5t^{2}+0.8t=2.28
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.
5t^{2}+0.8t-2.28=2.28-2.28
Subtract 2.28 from both sides of the equation.
5t^{2}+0.8t-2.28=0
Subtracting 2.28 from itself leaves 0.
t=\frac{-0.8±\sqrt{0.8^{2}-4\times 5\left(-2.28\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 0.8 for b, and -2.28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-0.8±\sqrt{0.64-4\times 5\left(-2.28\right)}}{2\times 5}
Square 0.8 by squaring both the numerator and the denominator of the fraction.
t=\frac{-0.8±\sqrt{0.64-20\left(-2.28\right)}}{2\times 5}
Multiply -4 times 5.
t=\frac{-0.8±\sqrt{0.64+45.6}}{2\times 5}
Multiply -20 times -2.28.
t=\frac{-0.8±\sqrt{46.24}}{2\times 5}
Add 0.64 to 45.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
t=\frac{-0.8±\frac{34}{5}}{2\times 5}
Take the square root of 46.24.
t=\frac{-0.8±\frac{34}{5}}{10}
Multiply 2 times 5.
t=\frac{6}{10}
Now solve the equation t=\frac{-0.8±\frac{34}{5}}{10} when ± is plus. Add -0.8 to \frac{34}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
t=\frac{3}{5}
Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.
t=-\frac{\frac{38}{5}}{10}
Now solve the equation t=\frac{-0.8±\frac{34}{5}}{10} when ± is minus. Subtract \frac{34}{5} from -0.8 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
t=-\frac{19}{25}
Divide -\frac{38}{5} by 10.
t=\frac{3}{5} t=-\frac{19}{25}
The equation is now solved.
5t^{2}+0.8t=2.28
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{5t^{2}+0.8t}{5}=\frac{2.28}{5}
Divide both sides by 5.
t^{2}+\frac{0.8}{5}t=\frac{2.28}{5}
Dividing by 5 undoes the multiplication by 5.
t^{2}+0.16t=\frac{2.28}{5}
Divide 0.8 by 5.
t^{2}+0.16t=0.456
Divide 2.28 by 5.
t^{2}+0.16t+0.08^{2}=0.456+0.08^{2}
Divide 0.16, the coefficient of the x term, by 2 to get 0.08. Then add the square of 0.08 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}+0.16t+0.0064=0.456+0.0064
Square 0.08 by squaring both the numerator and the denominator of the fraction.
t^{2}+0.16t+0.0064=0.4624
Add 0.456 to 0.0064 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t+0.08\right)^{2}=0.4624
Factor t^{2}+0.16t+0.0064. 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+0.08\right)^{2}}=\sqrt{0.4624}
Take the square root of both sides of the equation.
t+0.08=\frac{17}{25} t+0.08=-\frac{17}{25}
Simplify.
t=\frac{3}{5} t=-\frac{19}{25}
Subtract 0.08 from both sides of the equation.