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x^{2}-3x+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(-3\right)±\sqrt{\left(-3\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(-3\right)±\sqrt{9-4}}{2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{5}}{2}
Add 9 to -4.
x=\frac{3±\sqrt{5}}{2}
The opposite of -3 is 3.
x=\frac{\sqrt{5}+3}{2}
Now solve the equation x=\frac{3±\sqrt{5}}{2} when ± is plus. Add 3 to \sqrt{5}.
x=\frac{3-\sqrt{5}}{2}
Now solve the equation x=\frac{3±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from 3.
x^{2}-3x+1=\left(x-\frac{\sqrt{5}+3}{2}\right)\left(x-\frac{3-\sqrt{5}}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{5}}{2} for x_{1} and \frac{3-\sqrt{5}}{2} for x_{2}.