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

Question

3
𝑥
2
=
16
𝑥
+
12
. Which of the following statements gives the correct explanation?(1 point)
Responses

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

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.

Answer 1

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

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

Next, we identify the coefficients:

• \( a = 3 \)
• \( b = -16 \)
• \( c = -12 \)

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated using the formula:

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

Now, we calculate the discriminant:

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

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

Therefore, the correct explanation is: There will be two real solutions since the discriminant is positive.