SOLUTION: Given the equation {{{4x^2+8x+k=0}}} , for what values of k will the equation have a repeated root, 2 real roots, and 3 values of k that will give the equation rational roots. I un

Click here to see ALL problems on Linear Equations And Systems Word Problems

Question 997637: Given the equation , for what values of k will the equation have a repeated root, 2 real roots, and 3 values of k that will give the equation rational roots. I understand what it means to have those types of roots, I just don't understand how you would get to those specific answers. Like, I know how to find and imaginary root for this equation: . Would that be right?
Found 3 solutions by josgarithmetic, Boreal, MathLover1:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!
The discriminant of your equation is .
Simplified, .

Set discriminant to 0 to find what k gives two repeated roots. Solve for k.

Set discriminant to be greater than 0 to find what k gives two unique roots. Solve for k.

--
This, , did not render because bunch of notation wrong......
In pure text, it must read -infinity lessThan k lessThan 0,... unfortunately this will fail if done on this site page in the proper pure text symbolism, but the proper symbolism WILL render: .
To see that, click the View Source or Show Source link.

Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website!
4x^2+8x+k=0
4x^2+8x=-k
Factor out 4.
4(x^2+2x)= -k but k can be any constant, so it doesn't have to be divided by 4.
If you complete the square, the roots will be the same, -1.
4(x^2+2x+1)= -(k)+4, adding 4 to both sides
4(x+1)^2= -k +4
The right side has to equal zero for completing the square to give two roots of -1
-k=-4
k=4
4x^2+8x+4=0

========
Two real roots can be many things, so long as the discriminant b^2-4ac is positive.
b^2>4ak, since k is acting as c
64>16k
k<4 will work

=========
Rational roots work if the discriminant gives a perfect square, for all parts of the quadratic formula will be a fraction with nothing but integers.
b^2-16k has to be a perfect square
64-16k
k=0 works and roots are 4 and 0
k=3 :(1/8){-8 +/- sqrt(16){=(1/8)(-4) and (1/8)(-12)
k= -36/16 (1/8) (-8 +/- sqrt (100) and (1/8)(-8+10), (1/8) (-8-10) or (10/8), (-18/8)

Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website!

Given the equation , for what values of will the equation have a root (one double root), real roots, and values of k that will give the equation rational roots.
First of all, every polynomial has a discriminant, not just quadratics. The discriminant is the first place you look to classify the types of roots a polynomial has.
For example, a quadratic equation ax^2 + bx + c = 0, has a discriminant
If , there are real roots.
If there are two imaginary roots .
If , there is repeated root.

so, in your case we have
where , , and
If , there are real roots.
so,

or

take first number less then : and check the roots
...when you check it, you will find out that roots are: and

If there are two imaginary roots .

take and check

If , there is repeated root.

take and check