SOLUTION: Find k such that the equation 6x^2 + 6x + k =0 has Two equal real roots, two distinct real roots, and no real roots.

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Question 1085127: Find k such that the equation 6x^2 + 6x + k =0 has Two equal real roots, two distinct real roots, and no real roots.
Found 3 solutions by MathLover1, josmiceli, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!


6x%5E2+%2B+6x+%2B+k+=0->a=6, b=6, and+c=k
if the discriminant b%5E2-4ac%3E0, the equation has two distinct real roots
6%5E2-4%2A6k%3E0
36-24k%3E0
36%3E24k
k%3C36%2F24
k%3C6%2F4
k%3C3%2F2
k%3C1.5
so, we can try k=1 which is less than k%3C1.5
6x%5E2+%2B+6x+%2B+1+=0
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+6x%5E2+%2B+6x+%2B+1%29+


if the discriminant+b%5E2-4ac=0, the equation has two equal real roots
6%5E2-4%2A6k=0
36-24k=0
36=24k
k=36%2F24
k=6%2F4
k=3%2F2
k=1.5
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+6x%5E2+%2B+6x+%2B+1.5%29+

if the discriminant b%5E2-4ac%3C0, there are no real solutions
6%5E2-4%2A6k%3C0
36-24k%3C0
36%3C24k
k%3E36%2F24
k%3E6%2F4
k%3E3%2F2
k%3E1.5
if k=2->
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+6x%5E2+%2B+6x+%2B+2%29+

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+6x%5E2+%2B+6x+%2B+k+=+0+
Use quadratic formula
+x+=+%28+-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+6+
+b+=+6+
+c=+k+
----------------
+x+=+%28+-6+%2B-+sqrt%28+6%5E2+-+4%2A6%2Ak+%29%29+%2F+%282%2A6%29+
+x+=+%28+-6+%2B-+sqrt%28+36+-+24k+%29%29+%2F+12+
-------------------------------------------
(a)
2 equal real roots
+36+-+24k+ must be zero.
+36+-+24k+=+0+
+24k+=+36+
+k+=+3%2F2+
The 2 equal real roots are:
+x+=+-1%2F2+
----------------------
(b)
2 distinct real roots
+36+-+24k+ must be positive
I'll say
+36+-+24k+=+36+
+k+=+0+
The roots are:
+x+=+0+
+x+=+-1+
------------------------
No real roots
+36+-+24k+ must be negative
I'll say +36+-+24k+=+-36+
+24k+=+72+
+k+=+3+
The roots are:
+-1%2F2+%2B+%281%2F2%29%2Ai+
+-1%2F2+-+%281%2F2%29%2Ai+
-----------------------


Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Find k such that the equation 6x^2 + 6x + k =0 has Two equal real roots, two distinct real roots, and no real roots.