Wolfram|Alpha offers ns $-frac14 (1+sqrt5-2 x) (-1+sqrt5+2 x)$.Cystartupcuba.org gives me $(x-frac1+sqrt52)(x-frac1-sqrt52)$.

You are watching: Factor x^2-x+1

ns closest i can gain ins $(x+1)(x-1)-x$.

for this reason how execute i obtain a nice answer choose the onens listed above?

finish ns square: Gatshe $x^2-x$ and also whatever before constant girlfriend must develop something that the develop $(x-c)^2$, then repair ns transforms you"ve made:$$ extStyle x^2-x-1 = left( x^2-x+frac14 ight) - frac14-1 = (x-frac12)^2-frac54$$currently the RHs has actually ns create $a^2-b^2$ i m sorry you have the right to aspect as $(a+b)(a-b)$:$$ extStyle (x-frac12)^2-frac54 = (x-frac12)^2-(fracsqrt52)^2=(x-frac12+fracsqrt52)(x-frac12-fracsqrt52)$$

Equipment 1: If $p$ is a root of $f(x)=x^2-x-1$ climate $x-p$ ins a element the $f$ (view https://en.wikipedia.org/wiki/Polynomial_remainder_theorem). For this reason $f(x)=(x-a)(x-b)$ wright here $a,b$ to be the root that $f$ (offered by the quadratic formula). This offers Cystartupcuba.org"s answer.

If you clear ns platform in Cystartupcuba.org"ns answer, you obtain Wolfram"s answer.

Equipment 2: finish the square. $x^2-x-1\=x^2-x+1/4-1-1/4 \= (x-1/2)^2-5/4$,

i beg your pardon is a difference of squares, for this reason ins factors together $(x-1/2-sqrns 5/2)(x-1/2+sqrt5/2)$. Thins is Cystartupcuba.org"ns answer.


apply quadratic formula because that ns roots that $x^2-x-1=0$ together follows $$x=frac-(-1)pmsqrt(-1)^2-4(1)(-1)2(1)=frac1afternoon sqrns 52$$hence, one should have actually ns following determinants $$x^2-x-1=1cdot left(x-frac1+ sqrns 52 ight)left(x-frac1- sqrns 52 ight)$$or $$frac14(2x-1-sqrt 5)(2x-1+sqrns 5)$$$$=-frac14(1+sqrns 5-2x)(-1+sqrns 5+2x)$$so both ns answers are correct


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