We begin by factoring the perfect square trinomial and solving the equation by square roots.

Step 1: Recognize and factor the perfect square trinomial

The given equation is:

$x^2 - 14x + 49 = 4$

The left-hand side $x^2 - 14x + 49$ is a perfect square trinomial because:

$x^2 - 14x + 49 = (x - 7)^2$

So the equation becomes:

$(x - 7)^2 = 4$

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Step 2: Solve by square roots

Take the square root of both sides:

$\sqrt{(x - 7)^2} = \pm \sqrt{4}$

$x - 7 = \pm 2$

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Step 3: Solve for $x$

Split into two cases:

1. $x - 7 = 2$ $x = 7 + 2 = 9$

2. $x - 7 = -2$ $x = 7 - 2 = 5$

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Final Answer:

The solutions are:

$x = 9 \quad \text{or} \quad x = 5$

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Do you need further explanation, or should I expand on any specific step?

Related Questions:

1. What are the properties of a perfect square trinomial?
2. How do you identify perfect square trinomials in general?
3. Why do we include $\pm$ when solving square root equations?
4. Can you explain the geometric interpretation of solving $(x - 7)^2 = 4$?
5. What happens if the right-hand side of the equation is negative?

Tip:

Always verify your solutions by substituting them back into the original equation.