To solve the equation $x^2 - 4x = 5$ algebraically, follow these steps:

Step 1: Rearrange the equation

Move all terms to one side of the equation to set it equal to zero:

$x^2 - 4x - 5 = 0$

Now, we have a quadratic equation in standard form.

Step 2: Factor the quadratic equation

We need to factor the quadratic expression $x^2 - 4x - 5$. We are looking for two numbers that multiply to $-5$ (the constant term) and add up to $-4$ (the coefficient of $x$).

The numbers that satisfy these conditions are $-5$ and $1$, because:

$-5 \times 1 = -5 \quad \text{and} \quad -5 + 1 = -4$

So, we can factor the quadratic expression as:

$(x - 5)(x + 1) = 0$

Step 3: Solve for $x$

Now, apply the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero:

$x - 5 = 0 \quad \text{or} \quad x + 1 = 0$

Solving each equation gives:

$x = 5 \quad \text{or} \quad x = -1$

Final Answer:

The solutions are: $x = 5 \quad \text{or} \quad x = -1$

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Would you like a more detailed breakdown of any step or have any questions about this process? Here are some related questions for further exploration:

1. What is the quadratic formula, and how can it be used to solve quadratic equations?
2. How do you complete the square to solve a quadratic equation?
3. What are the properties of quadratic functions and their graphs?
4. How do you factor quadratics when the leading coefficient is not 1?
5. What is the discriminant, and how does it determine the nature of the roots of a quadratic equation?

Tip: When factoring quadratics, always check the signs of the numbers you're factoring—positive or negative—because this will affect your solution!