Question 181636
{{{6x^2 - 5x + 1 = 0}}}
You can work backwards to get the equation
from the roots, which are:
(1) {{{x = 1/3}}}
(2) {{{x = 1/2}}}
Subtract {{{1/3}}} from both sides of (1), and
Subtract {{{1/2}}} from both sides of (2)
(1) {{{x - (1/3) = 0}}}
(2) {{{x - (1/2) = 0}}}
Now multiply both sides of (1) by {{{3}}},
and multiply both sides of (2) by {{{2}}}
(1) {{{3x - 1 = 0}}}
(2) {{{2x - 1 = 0}}}
The product of (1) and (2) will be {{{0}}} also
{{{(3x - 1)(2x - 1) = 6x^2 - 5x + 1}}}
{{{6x^2 - 5x + 1 = 0}}}
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Also, each of the roots will solve the equation
{{{6*(1/3)^2 - 5*(1/3) + 1 = 0}}}
{{{6*(1/9) - 5/3 + 1 = 0}}}
{{{6*(1/9) - 15/9 + 9/9 = 0}}}
Multiply both sides by {{{9}}}
{{{6 - 15 + 9 = 0}}}
{{{0 = 0}}}
And, also
{{{6*(1/2)^2 - 5*(1/2) + 1 = 0}}}
{{{6/4 - 5/2 + 1 = 0}}}
{{{6/4 - 10/4 + 4/4 = 0}}}
Multiply both sides by {{{4}}}
{{{6 - 10 + 4 = 0}}}
{{{0 = 0}}}
OK