Use the discriminant to determine the number of real solutions of 3x2=16x+12

Question

. Which of the following statements gives the correct explanation?(1 point)
Responses

There will be two real solutions since the discriminant is positive.
There will be two real solutions since the discriminant is positive.

There will be two real solutions since the discriminant is negative.
There will be two real solutions since the discriminant is negative.

There will be one real solution since the discriminant is zero.
There will be one real solution since the discriminant is zero.

There will be no real solutions since the discriminant is negative.

Answer 1

To determine the number of real solutions of the equation \(3x^2 = 16x + 12\), we can first rewrite it in the standard form of a quadratic equation:

\[ 3x^2 - 16x - 12 = 0 \]

In this equation, \(a = 3\), \(b = -16\), and \(c = -12\).

Next, we calculate the discriminant \(D\) using the formula:

\[ D = b^2 - 4ac \]

Substituting in the values of \(a\), \(b\), and \(c\):

\[ D = (-16)^2 - 4 \cdot 3 \cdot (-12) \] \[ D = 256 + 144 \] \[ D = 400 \]

Since the discriminant \(D = 400\) is positive, it indicates that there are two distinct real solutions.

Thus, the correct statement is:

There will be two real solutions since the discriminant is positive.