S and G are roots of x2−3x+1=0. Find equation whose roots are 1S−2 and 1G−2.

Note SG=1,&gt;S+G=3. Then1S−21G−2=1SG−2(S+G)+4=−11S−2+1G−2=S+G−4SG−2(S+G)+4=1Thus, the equation with roots 1S−2 and 1G−2 isx2−x−1=0

The substitution method your teacher is referring to goes as follows:Let y=1x−2 where x=S, G are the roots of the given equation x2−3x+1=0Rearranging gives x=1+2yySubstituting this into the equation satisfied by x gives(1+2yy)2−3(1+2yy)+1=0This simplifies to becomey2−y−1=0This is the same as the answer obtained by Quanto using Vieta's formulas.

Note SG=1,&gt;S+G=3. Then1S−21G−2=1SG−2(S+G)+4=−11S−2+1G−2=S+G−4SG−2(S+G)+4=1Thus, the equation with roots 1S−2 and 1G−2 isx2−x−1=0

The substitution method your teacher is referring to goes as follows:Let y=1x−2 where x=S, G are the roots of the given equation x2−3x+1=0Rearranging gives x=1+2yySubstituting this into the equation satisfied by x gives(1+2yy)2−3(1+2yy)+1=0This simplifies to becomey2−y−1=0This is the same as the answer obtained by Quanto using Vieta's formulas.

"S and G are roots of x2−3x+1=0. Find..." solved and explained

High School
Algebra
Algebra I
Quadratic function and equation

aanvarendbq28

Answered question

2022-03-30

S and G are roots of

x^{2} - 3 x + 1 = 0

. Find equation whose roots are

\frac{1}{S - 2}

and

\frac{1}{G - 2}

Answer & Explanation

Laylah Hebert

Beginner2022-03-31Added 15 answers

Note

S G = 1, > S + G = 3

. Then

\frac{1}{S - 2} \frac{1}{G - 2} = \frac{1}{S G - 2 (S + G) + 4} = - 1

\frac{1}{S - 2} + \frac{1}{G - 2} = \frac{S + G - 4}{S G - 2 (S + G) + 4} = 1

Thus, the equation with roots

\frac{1}{S - 2}

and

\frac{1}{G - 2}

x^{2} - x - 1 = 0

pypberissootcu

Beginner2022-04-01Added 14 answers

The substitution method your teacher is referring to goes as follows:
Let

y = \frac{1}{x - 2}

where

x = S

, G are the roots of the given equation

x^{2} - 3 x + 1 = 0

Rearranging gives

x = \frac{1 + 2 y}{y}

Substituting this into the equation satisfied by x gives

{(\frac{1 + 2 y}{y})}^{2} - 3 (\frac{1 + 2 y}{y}) + 1 = 0

This simplifies to become

y^{2} - y - 1 = 0

This is the same as the answer obtained by Quanto using Vieta's formulas.

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