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}}}

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> 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}}}       Log On


   

Question 709189: Find the values of m if the Quad. eqn. m%28x%5E2%2Bx%2B1%29%2Bx=x%5E2%2B1 has 2 real and equal roots. Hence, find the corresponding root of the equation based on each value of m that you have found.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
m%28x%5E2%2Bx%2B1%29%2Bx=x%5E2%2B1
First let's put the equation into standard ax%5E2+%2Bbx+%2Bc+=+0 form. We'll start by simplifying the left side:
mx%5E2%2Bmx%2Bm%2Bx=x%5E2%2B1
Then we'll gather all the terms on one side (so the other side is zero):
mx%5E2%2Bmx%2Bm%2Bx-x%5E2-1+=+0
Next we will gather and group the x%5E2 terms, the x terms and the "other terms":
%28mx%5E2-x%5E2%29%2B%28mx%2Bx%29%2B+%28m-1%29+=+0
Factoring out x%5E2 from the first group and the x from the second group we get:
x%5E2%28m-1%29%2Bx%28m%2B1%29%2B+%28m-1%29+=+0
To make this look more like standard form I will use the Commutative Property to switch the order of the factors:
%28m-1%29x%5E2%2B%28m%2B1%29x+%2B+%28m-1%29+=+0
We now have standard form with...
a = m-1
b = m+1
c = m-1

We will get equal roots if b%5E2-4ac (the discriminant) = 0. Replacing the a, b and c we found above into this equation we get:
%28m%2B1%29%5E2-4%28m-1%29%28m-1%29+=+0
Simplifying we get:
m%5E2%2B2m%2B1-4%28m-1%29%28m-1%29+=+0
m%5E2%2B2m%2B1-4%28m%5E2-2m%2B1%29+=+0
m%5E2%2B2m%2B1-4m%5E2%2B8m-4+=+0
-3m%5E2%2B10m-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%283m%5E2-10m+%2B+3%29+=+0
Then we can factor more:
-1%283m-1%29%28m-3%29+=+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.

To find the roots when m = 1/3:
m%28x%5E2%2Bx%2B1%29%2Bx=x%5E2%2B1
Replace the m with 1/3:
%281%2F3%29%28x%5E2%2Bx%2B1%29%2Bx=x%5E2%2B1
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%5E2%2Bx%2B1%2B3x=3x%5E2%2B3
Simplifying...
x%5E2%2B4x%2B1=3x%5E2%2B3
Making one side zero:
0=2x%5E2-4x%2B2
Factoring:
0=2%28x%5E2-2x%2B1%29
0=2%28x-1%29%5E2
Zero Product Property:
2 = 0 or x-1%29%5E2
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.