Solving Quadratic Equations by Factoring Activity Maze - Classful - Free Printable

Let's solve the "Solving Quadratic Equations by Factoring Maze" step-by-step.

We begin at the "START HERE!" square and solve each quadratic equation by factoring. Then, we use the solutions to determine which path to take to the next square until we reach "FINISHED!"

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🔹 Step 1: Start Here!

Equation:
$$
x^2 + 4x - 12 = 0
$$

Factor:
We need two numbers that multiply to $-12$ and add to $+4$.
These are $6$ and $-2$:
$$
(x + 6)(x - 2) = 0
$$

Set each factor equal to zero:
$$
x + 6 = 0 \Rightarrow x = -6 \\
x - 2 = 0 \Rightarrow x = 2
$$

✔ Solutions: $x = -6$, $x = 2$

Now look for the path labeled:
> $x = -6$
> $x = 2$

This leads to the next square:
$$
2x^2 + 13x + 6 = 0
$$

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🔹 Step 2: Solve $2x^2 + 13x + 6 = 0$

We need two numbers that multiply to $2 \times 6 = 12$ and add to $13$.
Those are $12$ and $1$:
$$
2x^2 + 12x + x + 6 = 0 \\
(2x^2 + 12x) + (x + 6) = 0 \\
2x(x + 6) + 1(x + 6) = 0 \\
(2x + 1)(x + 6) = 0
$$

Set each factor to zero:
$$
2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \\
x + 6 = 0 \Rightarrow x = -6
$$

✔ Solutions: $x = -\frac{1}{2}$, $x = -6$

Look for the path with:
> $x = -\frac{1}{2}$
> $x = -6$

This leads to:
$$
x^2 - 9x = 0
$$

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🔹 Step 3: Solve $x^2 - 9x = 0$

Factor out $x$:
$$
x(x - 9) = 0
$$

Set each factor to zero:
$$
x = 0 \quad \text{or} \quad x = 9
$$

✔ Solutions: $x = 0$, $x = 9$

Follow the path with:
> $x = 0$
> $x = 9$

This leads to:
$$
4x^2 = 12x - 5
$$

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🔹 Step 4: Solve $4x^2 = 12x - 5$

Bring all terms to one side:
$$
4x^2 - 12x + 5 = 0
$$

We need two numbers that multiply to $4 \times 5 = 20$ and add to $-12$.
These are $-10$ and $-2$:
$$
4x^2 - 10x - 2x + 5 = 0 \\
(4x^2 - 10x) - (2x - 5) = 0 \\
2x(2x - 5) -1(2x - 5) = 0 \\
(2x - 1)(2x - 5) = 0
$$

Set each factor to zero:
$$
2x - 1 = 0 \Rightarrow x = \frac{1}{2} \\
2x - 5 = 0 \Rightarrow x = \frac{5}{2}
$$

✔ Solutions: $x = \frac{1}{2}$, $x = \frac{5}{2}$

Look for the path with:
> $x = \frac{1}{2}$
> $x = \frac{5}{2}$

This leads to the "FINISHED!" square.

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✔ Final Answer: The correct path is:

1. $x^2 + 4x - 12 = 0$ → $x = -6$, $x = 2$
2. $2x^2 + 13x + 6 = 0$ → $x = -\frac{1}{2}$, $x = -6$
3. $x^2 - 9x = 0$ → $x = 0$, $x = 9$
4. $4x^2 = 12x - 5$ → $x = \frac{1}{2}$, $x = \frac{5}{2}$
5. FINISHED!

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🧩 Summary of Correct Path:

- Start Here!
- → $x = -6$, $x = 2$ → Next: $2x^2 + 13x + 6 = 0$
- → $x = -\frac{1}{2}$, $x = -6$ → Next: $x^2 - 9x = 0$
- → $x = 0$, $x = 9$ → Next: $4x^2 = 12x - 5$
- → $x = \frac{1}{2}$, $x = \frac{5}{2}$ → FINISHED!

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✔ You’ve solved the maze successfully! The correct route uses the answers as shown above. This activity reinforces factoring quadratics and checking solutions.