Question

Determine the zero(s) of the quadratic function y=2x2+10x+8

. Show all steps.

(2 points)
Step 1: Plug the equation into my calculator.

Step 2: I will look for the -intercepts, or where the graph crosses the x-axis.

Answer 1

Determine the vertex and axis of symmetry of y=−x2+8x+7

The vertex is (, ).

The axis is symmetry is ​​​​​​​.

Answer 2

Jillian is trying to factor this quadratic equation x^2+7x=−10. What property of equality must she use to set her equation equal to 0, so that she can factor correctly?(1 point)
Responses

Addition Property of Equality

Subtraction Property of Equality

Multiplication Property of Equality

Division Property of Equality

Answer 3

To determine the zero(s) of the quadratic function \( y = 2x^2 + 10x + 8 \), we can follow a systematic approach to find the values of \( x \) where \( y = 0 \). Here are the steps:

### Step 1: Set the equation to zero
We first set the quadratic equation equal to zero:
\[
2x^2 + 10x + 8 = 0
\]

### Step 2: Use the Quadratic Formula
We can solve for \( x \) using the Quadratic Formula:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]
where \( a = 2 \), \( b = 10 \), and \( c = 8 \).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = 10^2 = 100
\]
\[
4ac = 4 \cdot 2 \cdot 8 = 64
\]
Now, calculate the discriminant:
\[
b^2 - 4ac = 100 - 64 = 36
\]

### Step 4: Substitute into the Quadratic Formula
Since the discriminant is positive, we have two real solutions:
\[
x = \frac{{-10 \pm \sqrt{36}}}{2 \cdot 2}
\]

Calculating the square root of the discriminant:
\[
\sqrt{36} = 6
\]
Now, substitute this back into the formula:
\[
x = \frac{{-10 \pm 6}}{4}
\]

### Step 5: Find the two solutions
Now we calculate the two possible values for \( x \):
1. For \( x = \frac{{-10 + 6}}{4} \):
\[
x = \frac{{-4}}{4} = -1
\]
2. For \( x = \frac{{-10 - 6}}{4} \):
\[
x = \frac{{-16}}{4} = -4
\]

### Step 6: List the zeros
The zeros of the quadratic function \( y = 2x^2 + 10x + 8 \) are:
\[
x = -1 \quad \text{and} \quad x = -4
\]

### Conclusion
Thus, the x-intercepts, or zeros of the function, where the graph crosses the x-axis, are \( x = -1 \) and \( x = -4 \).

Answer 4

To determine the vertex and axis of symmetry of the quadratic function \( y = -x^2 + 8x + 7 \), we can use the following steps:

### Step 1: Identify the coefficients
The quadratic can be expressed in the standard form \( y = ax^2 + bx + c \), where:
- \( a = -1 \)
- \( b = 8 \)
- \( c = 7 \)

### Step 2: Find the x-coordinate of the vertex
The x-coordinate of the vertex for a quadratic function in standard form is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{8}{2 \cdot -1} = -\frac{8}{-2} = 4
\]

### Step 3: Find the y-coordinate of the vertex
To find the y-coordinate of the vertex, substitute \( x = 4 \) back into the original equation:
\[
y = - (4)^2 + 8(4) + 7
\]
Calculating it step by step:
\[
y = -16 + 32 + 7
\]
\[
y = 16 + 7 = 23
\]

### Step 4: Write the vertex
The vertex of the quadratic function is:
\[
(4, 23)
\]

### Step 5: Determine the axis of symmetry
The axis of symmetry for a parabola in standard form can be expressed as the equation:
\[
x = -\frac{b}{2a}
\]
From our earlier calculation, the axis of symmetry is:
\[
x = 4
\]

### Conclusion
Thus, the vertex of the quadratic function \( y = -x^2 + 8x + 7 \) is:
\[
\text{Vertex} = (4, 23)
\]
And the axis of symmetry is:
\[
\text{Axis of symmetry} = x = 4
\]

Answer 5

To set the quadratic equation \( x^2 + 7x = -10 \) equal to 0 so that Jillian can factor it correctly, she needs to move the constant term (-10) to the other side of the equation. This is done using the **Addition Property of Equality**, which states that if you add the same value to both sides of an equation, the two sides remain equal.

In this case, Jillian can add 10 to both sides of the equation:

\[
x^2 + 7x + 10 = 0
\]

Thus, the correct response is:

**Addition Property of Equality**