SOLUTION: Find the values of {{{m}}} if the Quad. eqn. {{{m(x^2+x+1)+x=x^2+1}}} has 2 real and equal roots. Hence, find the corresponding root of the equation based on each value of {{{m}}}

Question 709189: Find the values of if the Quad. eqn. has 2 real and equal roots. Hence, find the corresponding root of the equation based on each value of that you have found.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First let's put the equation into standard form. We'll start by simplifying the left side:

Then we'll gather all the terms on one side (so the other side is zero):

Next we will gather and group the terms, the x terms and the "other terms":

Factoring out from the first group and the x from the second group we get:

To make this look more like standard form I will use the Commutative Property to switch the order of the factors:

We now have standard form with...
a = m-1
b = m+1
c = m-1

We will get equal roots if (the discriminant) = 0. Replacing the a, b and c we found above into this equation we get:

Simplifying we get:

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:

Then we can factor more:

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.

To find the roots when m = 1/3:

Replace the m with 1/3:

Now we solve for x. To make things easier I'm going to multiply each side by three to get rid of the fraction:

Simplifying...

Making one side zero:

Factoring:

Zero Product Property:
2 = 0 or
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.

I'll leave it up to you to figure out the equal roots when m = 3.
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