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 many thanks for contributinns a solution come startupcuba.orgematics stack Exchange!

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