Solving Quadratic Equations By Graphing Worksheet, Practice, And ... - Free Printable

You're correct — the worksheet asks you to solve quadratic equations by graphing and then use the graph to find the zeros (also called roots or x-intercepts).

Since I can’t draw graphs here, I’ll explain how to graph each quadratic function and then find its zeros from the graph, step-by-step. I’ll also give you the actual zeros so you can verify your graphs.

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## 🔍 Key Concept:
To solve a quadratic equation like `x² - 5x + 6 = 0` by graphing:
1. Rewrite it as a function: `y = x² - 5x + 6`
2. Graph the parabola.
3. The zeros are the x-values where the graph crosses the x-axis (i.e., where y = 0).

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# 📌 Problem 1: `x² - 5x + 6 = 0`

Step 1: Identify key features

- Standard form: `y = x² - 5x + 6`
- Opens upward (since coefficient of x² is positive)
- Vertex: Use formula `x = -b/(2a)` → `x = 5/2 = 2.5`
- Plug into equation: `y = (2.5)² - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25`
- Vertex: `(2.5, -0.25)`
- Y-intercept: When x=0 → `y = 6` → Point `(0, 6)`
- X-intercepts (zeros): Factor the equation!

Step 2: Factor to find zeros

`x² - 5x + 6 = (x - 2)(x - 3) = 0`
→ Zeros: x = 2 and x = 3

✔ So on your graph, the parabola should cross the x-axis at (2, 0) and (3, 0).

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# 📌 Problem 2: `x² - 9x + 20 = 0`

Step 1: Function: `y = x² - 9x + 20`

- Opens upward
- Vertex: `x = 9/2 = 4.5`
- `y = (4.5)² - 9(4.5) + 20 = 20.25 - 40.5 + 20 = -0.25`
- Vertex: `(4.5, -0.25)`
- Y-intercept: `(0, 20)`

Step 2: Factor

`x² - 9x + 20 = (x - 4)(x - 5) = 0`
→ Zeros: x = 4 and x = 5

✔ Graph crosses x-axis at (4, 0) and (5, 0).

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# 📌 Problem 3: `x² - 7x + 12 = 0`

Step 1: Function: `y = x² - 7x + 12`

- Opens upward
- Vertex: `x = 7/2 = 3.5`
- `y = (3.5)² - 7(3.5) + 12 = 12.25 - 24.5 + 12 = -0.25`
- Vertex: `(3.5, -0.25)`
- Y-intercept: `(0, 12)`

Step 2: Factor

`x² - 7x + 12 = (x - 3)(x - 4) = 0`
→ Zeros: x = 3 and x = 4

✔ Graph crosses x-axis at (3, 0) and (4, 0).

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# 📌 Problem 4: `x² + 6x + 9 = 0`

Step 1: Function: `y = x² + 6x + 9`

- Opens upward
- Vertex: `x = -6/2 = -3`
- `y = (-3)² + 6(-3) + 9 = 9 - 18 + 9 = 0`
- Vertex: `(-3, 0)` ← This is also the zero!
- Y-intercept: `(0, 9)`

Step 2: Factor

`x² + 6x + 9 = (x + 3)² = 0`
→ Zero: x = -3 (repeated root)

✔ Graph touches the x-axis at (-3, 0) — it’s a “tangent” point (vertex on x-axis).

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## ✔ Summary of Zeros:

| Problem | Equation | Zeros (x-intercepts) |
|---------|----------------------|------------------------------|
| 1 | x² - 5x + 6 = 0 | x = 2, x = 3 |
| 2 | x² - 9x + 20 = 0 | x = 4, x = 5 |
| 3 | x² - 7x + 12 = 0 | x = 3, x = 4 |
| 4 | x² + 6x + 9 = 0 | x = -3 (double root) |

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## 🖍️ How to Graph Each:

For each problem:

1. Draw your coordinate plane with labeled axes.
2. Plot the vertex.
3. Plot the y-intercept.
4. Plot the x-intercepts (zeros).
5. Sketch a smooth U-shaped curve (parabola) through these points.

💡 Tip: Since all these quadratics factor nicely, their graphs will have integer x-intercepts, making them easy to plot!

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## 🎯 Final Answer (Zeros Only):

1) x = 2, 3
2) x = 4, 5
3) x = 3, 4
4) x = -3

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Let me know if you want printable graph paper or step-by-step plotting instructions for any specific problem!