Question 709189
{{{m(x^2+x+1)+x=x^2+1}}}
First let's put the equation into standard {{{ax^2 +bx +c = 0}}} form. We'll start by simplifying the left side:
{{{mx^2+mx+m+x=x^2+1}}}
Then we'll gather all the terms on one side (so the other side is zero):
{{{mx^2+mx+m+x-x^2-1 = 0}}}
Next we will gather and group the {{{x^2}}} terms, the x terms and the "other terms":
{{{(mx^2-x^2)+(mx+x)+ (m-1) = 0}}}
Factoring out {{{x^2}}} from the first group and the x from the second group we get:
{{{x^2(m-1)+x(m+1)+ (m-1) = 0}}}
To make this look more like standard form I will use the Commutative Property to switch the order of the factors:
{{{(m-1)x^2+(m+1)x + (m-1) = 0}}}
We now have standard form with...
a = m-1
b = m+1
c = m-1<br>
We will get equal roots if {{{b^2-4ac}}} (the discriminant) = 0. Replacing the a, b and c we found above into this equation we get:
{{{(m+1)^2-4(m-1)(m-1) = 0}}}
Simplifying we get:
{{{m^2+2m+1-4(m-1)(m-1) = 0}}}
{{{m^2+2m+1-4(m^2-2m+1) = 0}}}
{{{m^2+2m+1-4m^2+8m-4 = 0}}}
{{{-3m^2+10m-3 = 0}}}
Now we solve for m. It will be easier to factor if we make the "a" in this quadratic positive. So we'll start by factoring out -1:
{{{-1(3m^2-10m + 3) = 0}}}
Then we can factor more:
{{{-1(3m-1)(m-3) = 0}}}
From the Zero Product Property:
-1 = 0 or 3m-1 = 0 or m-3 = 0
The first equation is false and has no solution. The other two equations have solutions:
m = 1/3 or m = 3
So if the m in your original equation is either 1/3 or 3 there will be two equal roots to the equation.<br>
To find the roots when m = 1/3:
{{{m(x^2+x+1)+x=x^2+1}}}
Replace the m with 1/3:
{{{(1/3)(x^2+x+1)+x=x^2+1}}}
Now we solve for x. To make things easier I'm going to multiply each side by three to get rid of the fraction:
{{{x^2+x+1+3x=3x^2+3}}}
Simplifying...
{{{x^2+4x+1=3x^2+3}}}
Making one side zero:
{{{0=2x^2-4x+2}}}
Factoring:
{{{0=2(x^2-2x+1)}}}
{{{0=2(x-1)^2}}}
Zero Product Property:
2 = 0 or {{{x-1)^2}}}
There is no solution to the first equation. But the second equation has a solution of x = 1. So when m = 1/3 your equation has two equal roots of 1.<br>
I'll leave it up to you to figure out the equal roots when m = 3.