To locate the zeros of the quadratic function \( y = -3x^2 + 12x + 15 \), we can set \( y \) to zero and solve for \( x \):
\[ -3x^2 + 12x + 15 = 0 \]
We can simplify this equation by multiplying everything by -1:
\[ 3x^2 - 12x - 15 = 0 \]
Next, we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 3 \), \( b = -12 \), and \( c = -15 \).
First, we'll calculate the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4(3)(-15) \] \[ = 144 + 180 = 324 \]
Now, since the discriminant is positive, we will have two real and distinct roots. Now, we'll compute the roots using the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{324}}{2(3)} \] \[ = \frac{12 \pm 18}{6} \]
Now, calculating the two potential solutions:
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For \( 12 + 18 \): \[ x = \frac{30}{6} = 5 \]
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For \( 12 - 18 \): \[ x = \frac{-6}{6} = -1 \]
Thus, the zeros of the function are \( x = 5 \) and \( x = -1 \).
So the response indicating the zeros is:
\((−1,0)(5,0)\).
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