Question

Use any method to locate the zeros of y=−3x2+12x+15 .(1 point)
Responses

There are no zeros.
There are no zeros.

(0,15)
left parenthesis 0 comma 15 right parenthesis

(−1,0)(5,0)

Answer 1

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:

1. For \( 12 + 18 \):
\[
x = \frac{30}{6} = 5
\]

2. 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:

\((−1,0)(5,0)\).