To determine which answer choice (if any) contains all the roots of the equation [tex]\( -12x^2 = 60x + 75 \)[/tex], let's proceed with the necessary algebraic steps:
1. Rewrite the equation in standard form:
The given equation is:
[tex]\[
-12x^2 = 60x + 75
\][/tex]
To get this into the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex], we move all terms to the left side of the equation:
[tex]\[
-12x^2 - 60x - 75 = 0
\][/tex]
2. Solve the quadratic equation:
We can solve this quadratic equation using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = -12 \)[/tex], [tex]\( b = -60 \)[/tex], and [tex]\( c = -75 \)[/tex].
Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[
\Delta = b^2 - 4ac = (-60)^2 - 4(-12)(-75)
\][/tex]
[tex]\[
\Delta = 3600 - 3600 = 0
\][/tex]
3. Determine the roots:
Since the discriminant is zero ([tex]\(\Delta = 0\)[/tex]), the quadratic equation has exactly one real double root (real and repeated root):
[tex]\[
x = \frac{-b}{2a} = \frac{-(-60)}{2(-12)} = \frac{60}{-24} = -2.5
\][/tex]
So, the root of the equation is:
[tex]\[
x = -2.5
\][/tex]
4. Match the roots to the answer choices:
The given answer choices are:
[tex]\[
\text{1. } x = 0, 3
\][/tex]
[tex]\[
\text{2. } x = \pm 2.5
\][/tex]
[tex]\[
\text{3. } x = 3, -2.5
\][/tex]
[tex]\[
\text{4. } x = -2.5
\][/tex]
We found that the only root is [tex]\( x = -2.5 \)[/tex]. Among the choices, only choice 4 ( [tex]\( x = -2.5 \)[/tex] ) matches the solution.
Hence, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
Note that none of the other answer choices include `-2.5` exclusively or correctly include additional extraneous roots.
