Features of Graphs of Quadratic Equations Practice

Question

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Question
Determine the zeros of the quadratic function y=−x2+16x−39
. (When listing the smaller zero, remember that negative values are always smaller than positive values.)(1 point)
smaller zero: x=
; greater zero: x=
Remaining Attempts : 3
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Answer 1

To find the zeros of the quadratic function \( y = -x^2 + 16x - 39 \), you can set the equation equal to zero and solve for \( x \):

\[ -x^2 + 16x - 39 = 0 \]

To make the calculations easier, you might want to multiply through by -1 to simplify:

\[ x^2 - 16x + 39 = 0 \]

Now, we can use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -16 \), and \( c = 39 \).

1. Calculate the discriminant \( b^2 - 4ac \): \[ (-16)^2 - 4 \cdot 1 \cdot 39 = 256 - 156 = 100 \]

2. Apply the quadratic formula: \[ x = \frac{-(-16) \pm \sqrt{100}}{2 \cdot 1} = \frac{16 \pm 10}{2} \]

3. Find the two potential solutions: \[ x = \frac{16 + 10}{2} = \frac{26}{2} = 13 \] \[ x = \frac{16 - 10}{2} = \frac{6}{2} = 3 \]

So, the smaller zero is \( x = 3 \) and the greater zero is \( x = 13 \).

Final Answer: smaller zero: \( x = 3 \); greater zero: \( x = 13 \)