To find the zeros of the quadratic function \( y = -3x^2 + 12x + 15 \), we set the equation equal to zero:
\[ -3x^2 + 12x + 15 = 0 \]
Next, we can either factor the quadratic, use the quadratic formula, or complete the square. Here, we will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, the coefficients are:
- \( a = -3 \)
- \( b = 12 \)
- \( c = 15 \)
We first calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = 12^2 = 144 \] \[ 4ac = 4 \cdot (-3) \cdot 15 = -180 \] \[ b^2 - 4ac = 144 - (-180) = 144 + 180 = 324 \]
Since the discriminant is positive (324), there are two distinct real roots.
Now we use the quadratic formula:
\[ x = \frac{-12 \pm \sqrt{324}}{2 \cdot -3} \]
Calculate \( \sqrt{324} = 18 \):
\[ x = \frac{-12 \pm 18}{-6} \]
Calculating the two possible solutions:
- \( x = \frac{-12 + 18}{-6} = \frac{6}{-6} = -1 \)
- \( x = \frac{-12 - 18}{-6} = \frac{-30}{-6} = 5 \)
Thus, the zeros of the function are at:
\[ (-1, 0) \quad \text{and} \quad (5, 0) \]
Now, we can match our result to the provided options:
- A. There are no zeros
- B. (1, 0) and (-5, 0)
- C. (-1, 0) and (5, 0)
- D. (0, 15)
The correct answer is C. (-1, 0) and (5, 0).
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