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x^{2}-9x+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{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4}}{2}
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(-9\right)±\sqrt{81-4}}{2}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{77}}{2}
Add 81 to -4.
x=\frac{9±\sqrt{77}}{2}
The opposite of -9 is 9.
x=\frac{\sqrt{77}+9}{2}
Now solve the equation x=\frac{9±\sqrt{77}}{2} when ± is plus. Add 9 to \sqrt{77}\approx 8.774964387.
x=\frac{9-\sqrt{77}}{2}
Now solve the equation x=\frac{9±\sqrt{77}}{2} when ± is minus. Subtract \sqrt{77}\approx 8.774964387 from 9.
x^{2}-9x+1=\left(x-\frac{\sqrt{77}+9}{2}\right)\left(x-\frac{9-\sqrt{77}}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{77}}{2}\approx 8.887482194 for x_{1} and \frac{9-\sqrt{77}}{2}\approx 0.112517806 for x_{2}.