4-2 Solving Quadratic Equations by Graphing - Free Printable

The image shows a worked example of solving a quadratic equation by graphing. Let's go through the problem step-by-step and explain the solution.

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Problem:

Solve the quadratic equation
$$
x^2 + 6x + 8 = 0
$$
by graphing.

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Step 1: Identify the Quadratic Function

We are given:
$$
f(x) = x^2 + 6x + 8
$$

We want to find the x-intercepts (zeros) of this function, which are the solutions to the equation $ f(x) = 0 $.

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Step 2: Find the Vertex

For a quadratic in the form $ f(x) = ax^2 + bx + c $, the x-coordinate of the vertex is:
$$
x = -\frac{b}{2a}
$$

Here, $ a = 1 $, $ b = 6 $, so:
$$
x = -\frac{6}{2(1)} = -3
$$

Now plug $ x = -3 $ into the function to find the y-coordinate:
$$
f(-3) = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1
$$

So, the vertex is at:
$$
(-3, -1)
$$

This is the lowest point of the parabola since $ a > 0 $ (opens upward).

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Step 3: Create a Table of Values

We calculate $ y = f(x) $ for several values of $ x $ around the vertex:

| $ x $ | $ y = x^2 + 6x + 8 $ |
|--------|------------------------|
| -5 | $ 25 - 30 + 8 = 3 $ |
| -4 | $ 16 - 24 + 8 = 0 $ |
| -3 | $ 9 - 18 + 8 = -1 $ |
| -2 | $ 4 - 12 + 8 = 0 $ |
| -1 | $ 1 - 6 + 8 = 3 $ |

So we have points:
- $ (-5, 3) $
- $ (-4, 0) $
- $ (-3, -1) $
- $ (-2, 0) $
- $ (-1, 3) $

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Step 4: Graph the Parabola

Plot these points on the coordinate plane. The graph is a U-shaped parabola with vertex at $ (-3, -1) $. It passes through:
- $ (-4, 0) $
- $ (-2, 0) $

These two points are where the graph crosses the x-axis, meaning they are the solutions to the equation.

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Step 5: Find the Solutions

The x-intercepts are the solutions to $ x^2 + 6x + 8 = 0 $.

From the graph and table:
- $ x = -4 $
- $ x = -2 $

So, the solutions are:
$$
\boxed{x = -4 \quad \text{and} \quad x = -2}
$$

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✔ Final Answer:

$$
\boxed{x = -4 \quad \text{or} \quad x = -2}
$$

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Verification (Optional):

Factor the quadratic:
$$
x^2 + 6x + 8 = (x + 4)(x + 2) = 0
$$

Set each factor to zero:
- $ x + 4 = 0 \Rightarrow x = -4 $
- $ x + 2 = 0 \Rightarrow x = -2 $

✔ Matches the graphical solution.

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Summary:

By graphing $ y = x^2 + 6x + 8 $, we found that it crosses the x-axis at $ x = -4 $ and $ x = -2 $. These are the solutions to the equation. The method works because the x-intercepts of the graph represent the roots of the quadratic equation.