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.

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

To determine the types of roots for each equation using the discriminant, you need to understand the formula and the conditions for each scenario.

The discriminant, denoted by Δ, is the part of the quadratic formula under the square root symbol. For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by:

Δ = b^2 - 4ac

Now let's apply this to the given equations:

1. x^2 - 4x = 5

To find the discriminant, we compare the equation with the standard quadratic form ax^2 + bx + c = 0. In this case, a = 1, b = -4, and c = -5. So, the discriminant is:

Δ = (-4)^2 - 4(1)(-5) = 16 + 20 = 36

Since the discriminant is positive (Δ > 0) and a perfect square, the equation has two rational roots. So the answer is (a) Two rational roots.

2. -3x^3 + 19x - 20 = 0

Again, compare the equation with the standard quadratic form. Here, a = -3, b = 19, and c = -20. The discriminant is:

Δ = (19)^2 - 4(-3)(-20) = 361 - 240 = 121

Since the discriminant is positive and a perfect square, the equation has two rational roots. So the answer is (a) Two rational roots.

3. -5x^2 - 7x - 5 = 0

In this equation, a = -5, b = -7, and c = -5. The discriminant is:

Δ = (-7)^2 - 4(-5)(-5) = 49 - 100 = -51

Since the discriminant is negative (Δ < 0), the equation has two complex roots. So the answer is (c) Two complex roots.

4. x^2 - 12 = 7x

Comparing this equation with the standard quadratic form, we have a = 1, b = -7, and c = -12. The discriminant is:

Δ = (-7)^2 - 4(1)(-12) = 49 + 48 = 97

Since the discriminant is positive but not a perfect square, the equation has two irrational roots. So the answer is (d) Two irrational roots.