I can't find the two distinct and equal roots,I ended up with answer than can't be solved...
Can someone use questions below as examples..
This is the formula:b^2-4ac=0
1.Find the values of k for which (k+1)x^2+kx-2k=0 has two equal roots.
2.........3x^2-4x+5-k=0 has two distinct roots.
go with oobleck on the both,
I read c = -k instead of c = -2k
I read c = -k instead of c = 5-k
time to get new reading glasses.
ok, you knew to use the discriminant. Too bad you didn't show your work. I hope you did
b^2 - 4ac = 0
k^2 - 4(-2k)(k+1) = 0
k(9k+8) = 0
It is clear that k=0 works
But we could have known that from the start:
y = (k+1)x^2+kx-2k
It's not quite so obvious that k = -8/9 also works. But to check it,
y = 1/9 x^2 - 8/9 x + 16/9
= 1/9 (x^2 - 8x + 16) = 1/9 (x-4)^2
which has two roots of x=4.
If k=0, y = x^2 which has two roots x=0
For the second one,
3x^2-4x+5-k=0
you need a positive discriminant.
16 - 4(3)(5-k) > 0
16 - 60 + 12k > 0
12k > 44
k > 11/3
Or, complete the square. You will need something of the form
y = a(x-h)^2 - n
3x^2-4x+5-k = 1/3 (3x-2)^2 + 11/3 - k
You can see that if k = 11/3 then y is a perfect square with two roots of x = 2/3
If k > 11/3, the graph dips below the x-axis, giving two distinct roots
Just sub in your values:
b^2 - 4ac = 0
k^2 - 4(k+1)(-k) = 0
k^2 + 4k^2 + 4k = 0
5k^2 + 4k = 0
k(5k+4) = 0
k = 0 or k = -4/5
2 , this time b^2 - 4ac > 0
16 - 4(3)(-k) > 0
16 + 12k > 0
k > -16/12
k > -4/3
Mr/Ms Reiny, the first questions c is 2k thought isn't it supposed to be
k^2 - 4(k+1)(-2k)
Ohh thanks
How do you make the -4(k+1)(-2k) into (9k+8)?
To find the values of k for which the given quadratic equations have specific roots, we need to consider the discriminant of the quadratic equation. The discriminant is the expression inside the square root in the quadratic formula, which is b^2 - 4ac.
1. For the equation (k+1)x^2 + kx - 2k = 0:
In this equation, a = (k+1), b = k, and c = -2k. To have two equal roots, the discriminant should be equal to zero:
b^2 - 4ac = k^2 - 4(k+1)(-2k) = 0
Simplifying the equation, we get:
k^2 + 8k^2 + 8k = 0
9k^2 + 8k = 0
k(9k + 8) = 0
We have two solutions for k:
k = 0
9k + 8 = 0 --> k = -8/9
Therefore, the values of k for which the equation has two equal roots are k = 0 and k = -8/9.
2. For the equation 3x^2 - 4x + 5 - k = 0:
In this equation, a = 3, b = -4, and c = 5 - k. To have two distinct roots, the discriminant should be greater than zero:
b^2 - 4ac > 0
Substituting the values, we get:
(-4)^2 - 4(3)(5 - k) > 0
16 - 12(5 - k) > 0
16 - 60 + 12k > 0
12k - 44 > 0
12k > 44
k > 44/12
k > 11/3
Therefore, the values of k for which the equation has two distinct roots are k > 11/3.