Question

I really need help!
i have to make three equations:One function, f(x), with two real rational solutions.
One function, g(x), with two real irrational solutions.
One function, h(x), with two complex solutions.

Answer 1

Do you know what the discriminant of a quadratic equation is?

Answer 2

no

Answer 3

If \[\large ax ^{2} + bx + c = 0\]is the quadratic equation then the discriminant D = \[\Large D = b^{2} - 4ac\]

If D is positive you will have two real roots
If D is zero you will have two equal roots
If D is negative you will have two complex roots

If D is positive and is a perfect square you will get two real rational roots
If D is positive and is not a perfect square you will get two real irrational roots

Answer 4

wait is it where the number is the same?

Answer 5

ok thank you

Answer 6

They want f(x) to have two real rational roots: That means D must be positive and a perfect square.
The easiest way is to pick two rational roots, say 4 and 6. Then (x - 4)(x - 6) = 0 because the roots can be factored out.
(x - 4)(x - 6) = x^2 - 4x - 6x + 24 = 0
x^2 - 10x + 24

So f(x) = x^2 - 10x + 24 is an example of a function with two rational toots: 4 and 6.
That answers the first part.

Answer 7

If you calculate D for above you will see it is a perfect square.
a = 1, b = -10 and c = 24
D = b^2 - 4ac = (-10)^2 - (4)(1)(24) = 100 - 96 = 4 which is a perfect square (square of 2) and so the roots (or the solutions) will be real and rational.

Answer 8

For g(x) they want two real irrational roots. And for that all we need to do is make the discriminant D not a perfect square. SO we can use the same a and b values from the previous question and just change c slightly.
a = 1, b = -10, c = 23 (previously we had 24)
D = b^2 - 4ac = (-10)^2 - 4(1)(23) = 100 - 92 = 8 and 8 is not a perfect square. So it will have two real irrational roots.

So g(x) = x^2 - 10x + 23

Answer 9

To have two complex roots: just change c slightly again to make the discriminant negative.
a = 1, b = -10, c = 26
D = (-10)^2 - (4)(1)(26) = 100 - 104 = -4
Discriminant is negative so two complex roots

h(x) = x^2 - 10x + 26

So the answers are:

f(x) = x^2 - 10x + 24
g(x) = x^2 - 10x + 23
h(x) = x^2 - 10x + 26

Answer 10

Thank you so much!

Answer 11

you are welcome.