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

Algebra ->  Quadratic Equations and Parabolas -> 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      Log On


   

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) About Me  (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 ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-1x%2B2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A2%2A2=-15.

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 - sqrt%28+15%29+=+3.87298334620742.

The solution is

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-1%2Ax%2B2+%29

Answer by drk(1908) About Me  (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
d+=+%28-4%29%5E2+-+4%2Ak%2Ak
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
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
based on our polynomial, we get
x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A2%2A%28-3%29+%29%29%2F%282%2A2%29+
which is
x+=+%281+%2B-+sqrt%28+25%29%29%2F%284%29+
or
x+=+%281+%2B-5%29%2F4
we have two real rational roots.