35. For what range of values of 𝑥 is 3𝑥2 − 4𝑥 −

Question

4 nagative?
A. − 2
3
< 𝑥 < 2
B. −2 < 𝑥 < 2
3
C. 𝑥 < − 2
3
, 𝑥 > 2
D. 𝑥 < −2, 𝑥 > 2
3

Answer 1

First, we need to find the roots of the quadratic equation 3𝑥2 − 4𝑥 − 4 by setting it equal to zero and solving for 𝑥:

3𝑥2 − 4𝑥 − 4 = 0

Using the quadratic formula, we get:

𝑥 = [4 ± √(4^2 − 4(3)(−4))] / (2(3))
𝑥 = [4 ± √52] / 6
𝑥 = [2 ± √13] / 3

So the roots are 𝑥 = [2 + √13] / 3 and 𝑥 = [2 − √13] / 3.

To find the range of values for which the function is negative, we can use test points from each of the intervals determined by the roots.

Let's evaluate the function at 𝑥 = 0, 𝑥 = [2 − √13] / 3, and 𝑥 = [2 + √13] / 3:

If 𝑥 = 0, then 3𝑥2 − 4𝑥 − 4 = -4, which is negative.
If [2 − √13] / 3 < 𝑥 < [2 + √13] / 3, then 3𝑥2 − 4𝑥 − 4 < 0, since this is the interval between the two roots where the function is decreasing.
Otherwise, if 𝑥 < [2 − √13] / 3 or 𝑥 > [2 + √13] / 3, then 3𝑥2 − 4𝑥 − 4 > 0, since the function is increasing outside of the interval between the two roots.

Therefore, the range of values of 𝑥 for which 3𝑥2 − 4𝑥 − 4 is negative is given by:

𝑥 < [2 − √13] / 3 or 𝑥 > [2 + √13] / 3

This corresponds to option (D).

Answer 2

3𝑥2 − 4𝑥 − 4 = 0
(3x+2)(x-2) = 0
the parabola opens up, so y < 0 for x in (-2/3 , 2)
So, D is the answer, but your calculations are all wrong.

Answer 3

You are correct. Thank you for pointing out the error in my calculations.

Firstly, factoring the quadratic equation, as you did, gives us:

3𝑥2 − 4𝑥 − 4 = 0
(3𝑥 + 2)(𝑥 − 2) = 0

Thus, the roots are 𝑥 = −2/3 and 𝑥 = 2.

Next, since the coefficient of 𝑥2 is positive, the parabola opens upwards. Therefore, the function is negative in the interval between the two roots, which is:

−2/3 < 𝑥 < 2.

Hence, the correct option is D: 𝑥 < −2, 𝑥 > 2/3.

Thank you for bringing this to my attention.