To factorize and solve the quadratic equation $x^2 - 4x - 32 = 0$, follow these steps:

Step 1: Factorize the quadratic equation

We look for two numbers that:

1. Multiply to give $-32$ (the constant term).
2. Add to give $-4$ (the coefficient of $x$).

The numbers are $-8$ and $4$ because: $-8 \times 4 = -32 \quad \text{and} \quad -8 + 4 = -4.$

So, we write: $x^2 - 4x - 32 = (x - 8)(x + 4).$

Step 2: Solve for $x$

Set each factor equal to zero: $x - 8 = 0 \quad \text{or} \quad x + 4 = 0.$

Solve for $x$: $x = 8 \quad \text{or} \quad x = -4.$

Final Answer:

The solutions are: $x = 8 \quad \text{and} \quad x = -4.$

Would you like a detailed explanation of factorization or verification of the solution?

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Related Questions:

1. How do you verify if the solutions satisfy the equation?
2. What is the graph of $y = x^2 - 4x - 32$ and where does it intersect the x-axis?
3. How do you solve quadratic equations using the quadratic formula?
4. Can you explain completing the square for the same equation?
5. What are real-world applications of solving quadratic equations?

Tip:

When factoring quadratic equations, always check the product and sum conditions to confirm your factors are correct.