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10x^{2}+x-1=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-1±\sqrt{1^{2}-4\times 10\left(-1\right)}}{2\times 10}
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{-1±\sqrt{1-4\times 10\left(-1\right)}}{2\times 10}
Square 1.
x=\frac{-1±\sqrt{1-40\left(-1\right)}}{2\times 10}
Multiply -4 times 10.
x=\frac{-1±\sqrt{1+40}}{2\times 10}
Multiply -40 times -1.
x=\frac{-1±\sqrt{41}}{2\times 10}
Add 1 to 40.
x=\frac{-1±\sqrt{41}}{20}
Multiply 2 times 10.
x=\frac{\sqrt{41}-1}{20}
Now solve the equation x=\frac{-1±\sqrt{41}}{20} when ± is plus. Add -1 to \sqrt{41}.
x=\frac{-\sqrt{41}-1}{20}
Now solve the equation x=\frac{-1±\sqrt{41}}{20} when ± is minus. Subtract \sqrt{41} from -1.
10x^{2}+x-1=10\left(x-\frac{\sqrt{41}-1}{20}\right)\left(x-\frac{-\sqrt{41}-1}{20}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{41}}{20} for x_{1} and \frac{-1-\sqrt{41}}{20} for x_{2}.