Applications with the Discriminant

Today in class Mr.K got us to do some questions on Applications with the Discriminant. Here are the questions and I'll go over them after:

1)For what values of k will the equation: 2x²+4x+(2-k-k²)=0 have exactly one root?
2)3x²-mx+3=0 a) For what value(s) of m will one root be double the other

b) For what value(s) of m will the roots not be real

1) Okay for question #1 we'll use the discriminant formula which is b²-4ac and in order to find k we need to know what the discriminant has to equal to ensure a 1 root parabola and the number is...0. So sub in a, b, and c into the Discriminant Formula

b²-4ac=0................................................Discriminant Formula
(4)²-4(2)(2-k-k²)=0......................And balance equation
16-8(2-k-k²)=0
16=8(2-k-k²)
2=2-k-k²
0=-k-k²
0=k+k²
0=2k
0=k

So after all that work we now know that k must equal to 0 in order to have the quadratic equation have only 1 root.

2)First we'll answer b) because it will be used to find a)

b)First we'll plug in the a,b, and c into the Discriminant Formula. We' ll set the equation to zero and solve for m. I'll show you:

m²-4(3)(3)=0
m²-36=0..............Notice that m²-36 is a difference of squares?
(m-6)(m+6)=0
m=6 m=-6

Now let's look at the values of m on the number line:
Explanation: If we pick a number less than -6 and more than 6 and sub it in for we'll get a positive result but we need a negative number in order to get no roots so the numbers we need are anything between -6 and 6. m=(-6,6)= No roots

a) Okay now for question a. What are the roots in the quadratic formula? They are:
x#1=-b-√(b²-4ac)..........x#2=-b+√(b²-4ac)
..............2a...............................2a

So the equation to find the roots for the questions in 2x#1(the lowest root)=x#2. Now with that in mind we sub in x#1 and x#2 with that equations above and sub in a,b, and c from the original equation and now we have.

2(m-√(m²-36)=m+√(m²-36)
...........6.............6

Now just solve for m and viola those are the roots.

Okay that's my third scribe and I hope this helps with some questions on Applications with the Discriminant. Homework for Mr.K's class is Exercise #18. The next Scribe will be...m@rk. Good night everybody.