Question 119093
Determine a quadratic equation with integer coefficients that has roots
{{{(-1 +-  4sqrt(2))/2}}}. Can someone please help explain this to me ?
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You have to know this fact:
If the quadratic equation:

x² + bx + c = 0

has roots r<sub>1</sub> and r<sub>2</sub>, then b = -(r<sub>1</sub> + r<sub>2</sub>) and c = r<sub>1</sub>r<sub>2</sub>

So if the quadratic equation:

x² + bx + c = 0

is to has roots r<sub>1</sub> = {{{(-1+4sqrt(2))/2}}} and r<sub>2</sub> = {{{(-1-4sqrt(2))/2}}},
then b = -(r<sub>1</sub> + r<sub>2</sub>) = {{{-((-1+4sqrt(2))/2 + (-1-4sqrt(2))/2)}}} =

{{{-((-1+4sqrt(2)-1-4sqrt(2))/2)}}} =

{{{-(-1-1)/2}}} = {{{-(-2)/2}}} = {{{-(-1)}}} = {{{1}}}

and c = r<sub>1</sub>r<sub>2</sub> =  {{{((-1+4sqrt(2))/2)((-1-4sqrt(2))/2)}}} = 
{{{(-1+4sqrt(2))(-1-4sqrt(2))/4}}} = {{{(1+4sqrt(2)-4sqrt(2)-16sqrt(4))/4}}} =
{{{(1-16(2))/4}}} = {{{(1-32)/4}}} = {{{(-31)/4}}} = {{{-31/4}}}

So the equation

x² + bx + c = 0

becomes

x² + 1x + {{{(-31/4)}}} = 0

But that doesn't have integer coefficients, so we clear of
fractions so it will:

So we multiply every term by LCD = 4

4x² + 4x + {{{4(-31/4)}}} = 0

4x² + 4x - 31 = 0 

Now that one has only integer coefficients.

Edwin</pre>