Question

I have a few problems I need help with and also do have multiple choice. If I can have an explanation of how to get the answer that would be great.

1. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
x^2-4x=5
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.

2.Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
-3x^3+19x-20=0
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.

3. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots.
-5x^2-7x-5=0
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.

4. Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots
x^2-12=7x
a.) Two rational roots
b.) one repreated root
c.) two complex roots
d.) two irrational roots.

Answer 1

All these are based on the following concept (should be in your text)

for ax^2 + bx + c = 0

if b^2 - 4ac < 0 there are two imaginary or complex roots
if b^2 - 4ac = 0 , there is one root, (one repeated root)
if b^2 - 4ac > 0 , there are two distinct and real roots

if for the last case, b^2 - 4ac is a perfect square, then the two real roots will be rational, otherwise they are irrational.

in each case, evaluate b^2 - 4ac and decide where it fits in
remember to arrange the equation in the proper form

I will do #4

x^2 - 12 = 7x
x^2 - 7x - 12 = 0 , a=1, b=-7 , c= -12
b^2 - 4ac
= 49 - 4(1)(-12) = +97
so we will have 2 irrational roots