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4.9x^{2}-27.952x+60=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.
x=\frac{-\left(-27.952\right)±\sqrt{\left(-27.952\right)^{2}-4\times 4.9\times 60}}{2\times 4.9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, -27.952 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-27.952\right)±\sqrt{781.314304-4\times 4.9\times 60}}{2\times 4.9}
Square -27.952 by squaring both the numerator and the denominator of the fraction.
x=\frac{-\left(-27.952\right)±\sqrt{781.314304-19.6\times 60}}{2\times 4.9}
Multiply -4 times 4.9.
x=\frac{-\left(-27.952\right)±\sqrt{781.314304-1176}}{2\times 4.9}
Multiply -19.6 times 60.
x=\frac{-\left(-27.952\right)±\sqrt{-394.685696}}{2\times 4.9}
Add 781.314304 to -1176.
x=\frac{-\left(-27.952\right)±\frac{2\sqrt{1541741}i}{125}}{2\times 4.9}
Take the square root of -394.685696.
x=\frac{27.952±\frac{2\sqrt{1541741}i}{125}}{2\times 4.9}
The opposite of -27.952 is 27.952.
x=\frac{27.952±\frac{2\sqrt{1541741}i}{125}}{9.8}
Multiply 2 times 4.9.
x=\frac{3494+2\sqrt{1541741}i}{9.8\times 125}
Now solve the equation x=\frac{27.952±\frac{2\sqrt{1541741}i}{125}}{9.8} when ± is plus. Add 27.952 to \frac{2i\sqrt{1541741}}{125}.
x=\frac{3494+2\sqrt{1541741}i}{1225}
Divide \frac{3494+2i\sqrt{1541741}}{125} by 9.8 by multiplying \frac{3494+2i\sqrt{1541741}}{125} by the reciprocal of 9.8.
x=\frac{-2\sqrt{1541741}i+3494}{9.8\times 125}
Now solve the equation x=\frac{27.952±\frac{2\sqrt{1541741}i}{125}}{9.8} when ± is minus. Subtract \frac{2i\sqrt{1541741}}{125} from 27.952.
x=\frac{-2\sqrt{1541741}i+3494}{1225}
Divide \frac{3494-2i\sqrt{1541741}}{125} by 9.8 by multiplying \frac{3494-2i\sqrt{1541741}}{125} by the reciprocal of 9.8.
x=\frac{3494+2\sqrt{1541741}i}{1225} x=\frac{-2\sqrt{1541741}i+3494}{1225}
The equation is now solved.
4.9x^{2}-27.952x+60=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.
4.9x^{2}-27.952x+60-60=-60
Subtract 60 from both sides of the equation.
4.9x^{2}-27.952x=-60
Subtracting 60 from itself leaves 0.
\frac{4.9x^{2}-27.952x}{4.9}=-\frac{60}{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.
x^{2}+\left(-\frac{27.952}{4.9}\right)x=-\frac{60}{4.9}
Dividing by 4.9 undoes the multiplication by 4.9.
x^{2}-\frac{6988}{1225}x=-\frac{60}{4.9}
Divide -27.952 by 4.9 by multiplying -27.952 by the reciprocal of 4.9.
x^{2}-\frac{6988}{1225}x=-\frac{600}{49}
Divide -60 by 4.9 by multiplying -60 by the reciprocal of 4.9.
x^{2}-\frac{6988}{1225}x+\left(-\frac{3494}{1225}\right)^{2}=-\frac{600}{49}+\left(-\frac{3494}{1225}\right)^{2}
Divide -\frac{6988}{1225}, the coefficient of the x term, by 2 to get -\frac{3494}{1225}. Then add the square of -\frac{3494}{1225} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{6988}{1225}x+\frac{12208036}{1500625}=-\frac{600}{49}+\frac{12208036}{1500625}
Square -\frac{3494}{1225} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{6988}{1225}x+\frac{12208036}{1500625}=-\frac{6166964}{1500625}
Add -\frac{600}{49} to \frac{12208036}{1500625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{3494}{1225}\right)^{2}=-\frac{6166964}{1500625}
Factor x^{2}-\frac{6988}{1225}x+\frac{12208036}{1500625}. 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(x-\frac{3494}{1225}\right)^{2}}=\sqrt{-\frac{6166964}{1500625}}
Take the square root of both sides of the equation.
x-\frac{3494}{1225}=\frac{2\sqrt{1541741}i}{1225} x-\frac{3494}{1225}=-\frac{2\sqrt{1541741}i}{1225}
Simplify.
x=\frac{3494+2\sqrt{1541741}i}{1225} x=\frac{-2\sqrt{1541741}i+3494}{1225}
Add \frac{3494}{1225} to both sides of the equation.