SOLUTION: a)What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ? b)Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots. c)What is

Question 259039: a)What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ?
b)Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots.
c)What is the nature of the roots of 2x^2 - x - 3 = -5?
I asked these together because they are all similar. Thanks for any help.
Found 2 solutions by richwmiller, drk:
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
a)
2x^2 - kx - 3=0
just plug in 3 for x and solve for k
2*3^2 -3k - 3=0
2*9-3k-3=0
15=3k
5=k
b)
b^2-4ac must =0
(-1)^2-4(k*k)=0
1=4k^2
1/4=k^2
k=+1/2 and -1/2
(-1)^2-4*(-1/2)(-1/2
1-1=0
also works with +1/2
c)
2x^2 - x - 3 = -5
2x^2-x+2=0
two complex numbers because the discriminant(b^2-4ac=(-1)^2-(4*2*(-3)) is less than zero

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

The discriminant -15 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

In the field of imaginary numbers, the square root of -15 is + or - .

The solution is

Here's your graph:

Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
Lets take one at a time:
a) What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ?
x = 3 is a root means that (x-3)=0.
Now, (x-3)(2x+1) = 2x^2 -5x -3, so k = 5.
---
b) Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots.
If a quadratic has two equal roots the discriminant, or D = b^2-4ac = 0.
Applying this to our function, we get

or
16-4k^2 = 0
16 = 4k^2
4 = k^2
K = +-2
--
c) What is the nature of the roots of 2x^2 - x - 3 = -5?
When they say nature of the roots, we are going after the quadratic formula which is

based on our polynomial, we get

which is

or

we have two real rational roots.
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