Question 1126754
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We want to put the given 4th degree polynomial equation in the following form:<br>
{{{(x^2 +ax)^2 +b(x^2 +ax) +c=0}}}<br>
Expand this form and combine like terms:<br>
{{{(x^4+2ax^3+a^2x^2)+(bx^2+abx) + c = 0}}}<br>
{{{x^4+(2a)x^3+(a^2+b)x^2+(ab)x+c = 0}}}<br>
We want this to be equal to<br>
{{{x^4 -6x^3 +7x^2 +6x-3=0}}}<br>
Solve for a, b, and c by equating coefficients.<br>
x^4: the coefficients are both 1<br>
x^3: {{{2a = -6}}}  -->  a = -3<br>
x^2: {{{a^2+b = (-3)^2+b = 9+b = 7}}}  -->  b = -2<br>
x^1: {{{ab = 6}}}  (we already know that; if the product of our a and b were NOT 6, then the roots could not be found by this method)<br>
x^0: c = -3<br>
The equation in the new form is then<br>
{{{(x^2-3x)^2-2(x^2-3x)-3 = 0}}}<br>
This is a quadratic equation with "x^2-3x" as the "variable".  Factor and solve.<br>
{{{((x^2-3x)-3)((x^2-3x)+1) = 0}}}<br>
{{{x^2-3x-3 = 0}}}  or  {{{x^2-3x+1 = 0}}}<br>
Neither of these quadratic expressions factors, so we get two pairs of irrational roots.<br>
{{{x = (3 +- sqrt(21))/2}}}  and  {{{x = (3 +- sqrt(5))/2}}}