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20x^{2}-27x-14=0
Subtract 14 from both sides.
a+b=-27 ab=20\left(-14\right)=-280
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 20x^{2}+ax+bx-14. To find a and b, set up a system to be solved.
1,-280 2,-140 4,-70 5,-56 7,-40 8,-35 10,-28 14,-20
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -280.
1-280=-279 2-140=-138 4-70=-66 5-56=-51 7-40=-33 8-35=-27 10-28=-18 14-20=-6
Calculate the sum for each pair.
a=-35 b=8
The solution is the pair that gives sum -27.
\left(20x^{2}-35x\right)+\left(8x-14\right)
Rewrite 20x^{2}-27x-14 as \left(20x^{2}-35x\right)+\left(8x-14\right).
5x\left(4x-7\right)+2\left(4x-7\right)
Factor out 5x in the first and 2 in the second group.
\left(4x-7\right)\left(5x+2\right)
Factor out common term 4x-7 by using distributive property.
x=\frac{7}{4} x=-\frac{2}{5}
To find equation solutions, solve 4x-7=0 and 5x+2=0.
20x^{2}-27x=14
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.
20x^{2}-27x-14=14-14
Subtract 14 from both sides of the equation.
20x^{2}-27x-14=0
Subtracting 14 from itself leaves 0.
x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 20\left(-14\right)}}{2\times 20}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -27 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-27\right)±\sqrt{729-4\times 20\left(-14\right)}}{2\times 20}
Square -27.
x=\frac{-\left(-27\right)±\sqrt{729-80\left(-14\right)}}{2\times 20}
Multiply -4 times 20.
x=\frac{-\left(-27\right)±\sqrt{729+1120}}{2\times 20}
Multiply -80 times -14.
x=\frac{-\left(-27\right)±\sqrt{1849}}{2\times 20}
Add 729 to 1120.
x=\frac{-\left(-27\right)±43}{2\times 20}
Take the square root of 1849.
x=\frac{27±43}{2\times 20}
The opposite of -27 is 27.
x=\frac{27±43}{40}
Multiply 2 times 20.
x=\frac{70}{40}
Now solve the equation x=\frac{27±43}{40} when ± is plus. Add 27 to 43.
x=\frac{7}{4}
Reduce the fraction \frac{70}{40} to lowest terms by extracting and canceling out 10.
x=-\frac{16}{40}
Now solve the equation x=\frac{27±43}{40} when ± is minus. Subtract 43 from 27.
x=-\frac{2}{5}
Reduce the fraction \frac{-16}{40} to lowest terms by extracting and canceling out 8.
x=\frac{7}{4} x=-\frac{2}{5}
The equation is now solved.
20x^{2}-27x=14
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{20x^{2}-27x}{20}=\frac{14}{20}
Divide both sides by 20.
x^{2}-\frac{27}{20}x=\frac{14}{20}
Dividing by 20 undoes the multiplication by 20.
x^{2}-\frac{27}{20}x=\frac{7}{10}
Reduce the fraction \frac{14}{20} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{27}{20}x+\left(-\frac{27}{40}\right)^{2}=\frac{7}{10}+\left(-\frac{27}{40}\right)^{2}
Divide -\frac{27}{20}, the coefficient of the x term, by 2 to get -\frac{27}{40}. Then add the square of -\frac{27}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{27}{20}x+\frac{729}{1600}=\frac{7}{10}+\frac{729}{1600}
Square -\frac{27}{40} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{27}{20}x+\frac{729}{1600}=\frac{1849}{1600}
Add \frac{7}{10} to \frac{729}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{27}{40}\right)^{2}=\frac{1849}{1600}
Factor x^{2}-\frac{27}{20}x+\frac{729}{1600}. 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{27}{40}\right)^{2}}=\sqrt{\frac{1849}{1600}}
Take the square root of both sides of the equation.
x-\frac{27}{40}=\frac{43}{40} x-\frac{27}{40}=-\frac{43}{40}
Simplify.
x=\frac{7}{4} x=-\frac{2}{5}
Add \frac{27}{40} to both sides of the equation.