Problem Analysis:
The task involves solving a quadratic equation and plotting its graph. The given quadratic equation is:
$$
2x^2 + 8x + 6 = 0
$$
The steps to solve the problem are as follows:
1.
Find the zeros of the quadratic equation.
2.
Create a table of values for the quadratic function.
3.
Plot the points from the table on a graph.
4.
Connect the points with a smooth curve (parabola).
Let's go through each step in detail.
---
Step 1: Find the Zeros of the Quadratic Equation
The zeros of the quadratic equation are the values of \( x \) for which \( y = 0 \). To find these, we solve:
$$
2x^2 + 8x + 6 = 0
$$
#### Simplify the Equation:
Divide the entire equation by 2 to simplify:
$$
x^2 + 4x + 3 = 0
$$
#### Factor the Quadratic Equation:
Factor the quadratic expression \( x^2 + 4x + 3 \):
$$
x^2 + 4x + 3 = (x + 1)(x + 3)
$$
Set each factor equal to zero:
$$
x + 1 = 0 \quad \text{or} \quad x + 3 = 0
$$
Solve for \( x \):
$$
x = -1 \quad \text{or} \quad x = -3
$$
Thus, the zeros of the quadratic equation are:
$$
x = -1 \quad \text{and} \quad x = -3
$$
---
Step 2: Create a Table of Values
To plot the quadratic function \( y = 2x^2 + 8x + 6 \), we need to create a table of \( (x, y) \) values. We will choose several \( x \)-values around the vertex and calculate the corresponding \( y \)-values.
#### Identify the Vertex:
The vertex form of a parabola \( ax^2 + bx + c \) has its vertex at:
$$
x = -\frac{b}{2a}
$$
For the given equation \( 2x^2 + 8x + 6 \):
- \( a = 2 \)
- \( b = 8 \)
Calculate the \( x \)-coordinate of the vertex:
$$
x = -\frac{8}{2 \cdot 2} = -\frac{8}{4} = -2
$$
Substitute \( x = -2 \) into the equation to find the \( y \)-coordinate of the vertex:
$$
y = 2(-2)^2 + 8(-2) + 6
$$
Simplify:
$$
y = 2(4) + 8(-2) + 6
$$
$$
y = 8 - 16 + 6
$$
$$
y = -2
$$
Thus, the vertex is at \( (-2, -2) \).
#### Create the Table:
Choose \( x \)-values symmetrically around the vertex \( x = -2 \). Let's use \( x = -4, -3, -2, -1, 0 \).
| \( x \) | \( y = 2x^2 + 8x + 6 \) |
|---------|--------------------------|
| \( -4 \) | \( 2(-4)^2 + 8(-4) + 6 = 2(16) - 32 + 6 = 32 - 32 + 6 = 6 \) |
| \( -3 \) | \( 2(-3)^2 + 8(-3) + 6 = 2(9) - 24 + 6 = 18 - 24 + 6 = 0 \) |
| \( -2 \) | \( 2(-2)^2 + 8(-2) + 6 = 2(4) - 16 + 6 = 8 - 16 + 6 = -2 \) |
| \( -1 \) | \( 2(-1)^2 + 8(-1) + 6 = 2(1) - 8 + 6 = 2 - 8 + 6 = 0 \) |
| \( 0 \) | \( 2(0)^2 + 8(0) + 6 = 0 + 0 + 6 = 6 \) |
The completed table is:
| \( x \) | \( y \) |
|---------|---------|
| \( -4 \) | \( 6 \) |
| \( -3 \) | \( 0 \) |
| \( -2 \) | \( -2 \) |
| \( -1 \) | \( 0 \) |
| \( 0 \) | \( 6 \) |
---
Step 3: Plot the Points and Connect Them
Plot the points from the table on a coordinate plane:
- \( (-4, 6) \)
- \( (-3, 0) \)
- \( (-2, -2) \)
- \( (-1, 0) \)
- \( (0, 6) \)
Connect these points with a smooth parabolic curve. The vertex \( (-2, -2) \) is the lowest point of the parabola since the coefficient of \( x^2 \) is positive (\( a = 2 \)), indicating an upward-opening parabola.
---
Final Answer:
The zeros of the quadratic equation are \( x = -1 \) and \( x = -3 \). The table of values is:
| \( x \) | \( y \) |
|---------|---------|
| \( -4 \) | \( 6 \) |
| \( -3 \) | \( 0 \) |
| \( -2 \) | \( -2 \) |
| \( -1 \) | \( 0 \) |
| \( 0 \) | \( 6 \) |
The graph is a parabola with the vertex at \( (-2, -2) \), opening upwards.
$$
\boxed{x = -1 \text{ and } x = -3}
$$
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations by graphing worksheet.
