Solve for x
x=\frac{1}{10}=0.1
x=-1
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10x^{2}+9x=1
Swap sides so that all variable terms are on the left hand side.
10x^{2}+9x-1=0
Subtract 1 from both sides.
a+b=9 ab=10\left(-1\right)=-10
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 10x^{2}+ax+bx-1. To find a and b, set up a system to be solved.
-1,10 -2,5
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.
-1+10=9 -2+5=3
Calculate the sum for each pair.
a=-1 b=10
The solution is the pair that gives sum 9.
\left(10x^{2}-x\right)+\left(10x-1\right)
Rewrite 10x^{2}+9x-1 as \left(10x^{2}-x\right)+\left(10x-1\right).
x\left(10x-1\right)+10x-1
Factor out x in 10x^{2}-x.
\left(10x-1\right)\left(x+1\right)
Factor out common term 10x-1 by using distributive property.
x=\frac{1}{10} x=-1
To find equation solutions, solve 10x-1=0 and x+1=0.
10x^{2}+9x=1
Swap sides so that all variable terms are on the left hand side.
10x^{2}+9x-1=0
Subtract 1 from both sides.
x=\frac{-9±\sqrt{9^{2}-4\times 10\left(-1\right)}}{2\times 10}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 9 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-9±\sqrt{81-4\times 10\left(-1\right)}}{2\times 10}
Square 9.
x=\frac{-9±\sqrt{81-40\left(-1\right)}}{2\times 10}
Multiply -4 times 10.
x=\frac{-9±\sqrt{81+40}}{2\times 10}
Multiply -40 times -1.
x=\frac{-9±\sqrt{121}}{2\times 10}
Add 81 to 40.
x=\frac{-9±11}{2\times 10}
Take the square root of 121.
x=\frac{-9±11}{20}
Multiply 2 times 10.
x=\frac{2}{20}
Now solve the equation x=\frac{-9±11}{20} when ± is plus. Add -9 to 11.
x=\frac{1}{10}
Reduce the fraction \frac{2}{20} to lowest terms by extracting and canceling out 2.
x=-\frac{20}{20}
Now solve the equation x=\frac{-9±11}{20} when ± is minus. Subtract 11 from -9.
x=-1
Divide -20 by 20.
x=\frac{1}{10} x=-1
The equation is now solved.
10x^{2}+9x=1
Swap sides so that all variable terms are on the left hand side.
\frac{10x^{2}+9x}{10}=\frac{1}{10}
Divide both sides by 10.
x^{2}+\frac{9}{10}x=\frac{1}{10}
Dividing by 10 undoes the multiplication by 10.
x^{2}+\frac{9}{10}x+\left(\frac{9}{20}\right)^{2}=\frac{1}{10}+\left(\frac{9}{20}\right)^{2}
Divide \frac{9}{10}, the coefficient of the x term, by 2 to get \frac{9}{20}. Then add the square of \frac{9}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{9}{10}x+\frac{81}{400}=\frac{1}{10}+\frac{81}{400}
Square \frac{9}{20} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{9}{10}x+\frac{81}{400}=\frac{121}{400}
Add \frac{1}{10} to \frac{81}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{9}{20}\right)^{2}=\frac{121}{400}
Factor x^{2}+\frac{9}{10}x+\frac{81}{400}. 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{9}{20}\right)^{2}}=\sqrt{\frac{121}{400}}
Take the square root of both sides of the equation.
x+\frac{9}{20}=\frac{11}{20} x+\frac{9}{20}=-\frac{11}{20}
Simplify.
x=\frac{1}{10} x=-1
Subtract \frac{9}{20} from both sides of the equation.
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