To determine which quadratic equations can be solved by extracting square roots, the key is identifying those that can be simplified into a form like $x^2 = k$, where $k$ is a constant, without the need for factoring or using the quadratic formula.

Let's analyze each equation:

a. $x^2 - 25 = 0$
This equation can be rewritten as $x^2 = 25$, so you can solve by taking square roots.
Thus, this equation can be solved by extracting square roots.

b. $x^2 + 4x + 1 = 0$
This is a full quadratic equation with a linear term ($4x$) and does not fit the form $x^2 = k$. Solving it requires the quadratic formula or factoring.
Thus, this equation cannot be solved by extracting square roots.

c. $x^2 + 3x = 0$
This equation can be factored as $x(x + 3) = 0$, but it cannot be directly solved by extracting square roots because of the linear term. Factoring or the quadratic formula is needed.
Thus, this equation cannot be solved by extracting square roots.

d. $4x^2 + 3x + 12 = 0$
This is also a full quadratic equation that includes both a linear term and a constant. It cannot be reduced to the form $x^2 = k$ for extracting square roots.
Thus, this equation cannot be solved by extracting square roots.

Answer: a. $x^2 - 25 = 0$ is the only one that can be solved by extracting square roots.

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Would you like any further details or clarifications?

Here are 5 related questions to further your understanding:

1. What is the general form of a quadratic equation?
2. How do you solve a quadratic equation using the quadratic formula?
3. What are the conditions for factoring a quadratic equation?
4. How can completing the square help in solving quadratic equations?
5. When can we use the method of extracting square roots for solving equations?

Tip: To solve by extracting square roots, the equation should have no linear term and should be reducible to the form $x^2 = k$.