Question

To open these doors, you must match the number and type of solutions for the following two functions in standard form.
f(x) = x2 + 6x – 16
g(x) = x2 +6x + 1
Match the following descriptions of the solutions to each of the functions above.
Hint: they each have their own match.

Two real irrationals solutions
Two real rationals solutions

Answer 1

Thanks, but can you explain why please

Answer 2

Are you familiar with the quadratic formula?

Answer 3

The Quadratic Formula

\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The radicand of the quadratic formula, \(b^2 - 4ac\), is called the discriminant.
The discriminant allows you to tell the nature of the solutions.

If a quadratic equation has real coefficients, then there are three possibilities with the discriminant and what you learn from it.

If the discriminant equals zero, then the quadratic equation has only one real solution.
If the discriminant is positive, then the quadratic equation has two different real solutions.
If the discriminant is negative, then the quadratic equation has two complex solutions that are a complex conjugates.

Answer 4

I kinda get the discriminant but when i tried to do the first on I got -28

Answer 5

Be careful with signs in the operations.

Here is the first function:
f(x) = x2 + 6x – 16

For the first function, we have a = 1, b = 6, and c = -16.

The discriminant is:
\(b^2 - 4ac = 6^2 - 4(1)(-16) = 36 - (-64) = 100\)

Answer 6

As you can see, the discriminant of the first equation is 100, a positive number.
Since the root of 100 is real, the roots are real.

Answer 7

Here is the second function:
\(f(x) = x^2 + 6x + 1\)

For the second function, we have a = 1, b = 6, and c = 1.

The discriminant is:
\(b^2 - 4ac = 6^2 - 4(1)(1) = \)

Can you calculate the discriminant above?

Answer 8

Notice that for the first equation, the discriminant is 100, which is positive.
That means the equation has real solutions.
In addition, the square root of 100 is 10, which is rational, so the first equation has real, rational solutions.

Now look at the discriminant of the second equation. The discriminant is 36 - 4 = 32. 32 is positive, so the solutions tot he equation are real.
32 is not a perfect square, so its square root is not a rational number. That means that the second equation has real, irrational solutions.

Answer 9

Thank you!!! Do you have time for another question?

Answer 10

Yes. One more.
Please start a new post.