Let's solve each of these questions step-by-step:

1. Which of the following quadratic equations is written in standard form?

A quadratic equation is written in standard form as ax² + bx + c = 0.

• A. $3x^2 = 4$: This is not in standard form because it's missing the zero on the right-hand side.
• B. $3x^2 - 4 = 0$: This is in standard form.
• C. $3x + 2x = -4$: This is not a quadratic equation, it's linear.
• D. $3x^2 + 2x - 4 = 0$: This is also in standard form.

Answer: B and D are both valid, but if you must choose one, B is the simpler standard form.

2. In the equation $5x^2 - 7 = 4x$, what is the value of $b$?

To find $b$, we first rearrange the equation into standard form $ax² + bx + c = 0$.

$5x^2 - 7 = 4x \implies 5x^2 - 4x - 7 = 0$

Here, $b = -4$.

Answer: B. -4

3. Which of these quadratic equations can be solved by extracting square roots?

An equation can be solved by extracting square roots if it is in the form $x^2 = k$ (where $k$ is a constant).

• A. $\frac{5}{2}x - \frac{x}{5} = 2$: This is not a quadratic equation.
• B. $3x - 2 = 3(x + 1)$: Not a quadratic equation, this simplifies to a linear equation.
• C. $x^2 - 49 = 0$: This can be solved by extracting square roots. $x^2 = 49 \implies x = \pm 7$.
• D. $x^2 + 4x + 4 = 0$: This can be solved by factoring, but not by extracting square roots.

Answer: C. $x^2 - 49 = 0$

4. For what values of $x$ will make the expression $3x^2 = 108$ true?

To solve for $x$, we divide both sides by 3:

$x^2 = \frac{108}{3} = 36$

Taking the square root of both sides:

$x = \pm 6$

Answer: B. $\pm 6$

5. With the given quadratic equation $x^2 - 11x + 24 = 0$, find the roots.

We can solve this by factoring:

$x^2 - 11x + 24 = 0 \implies (x - 3)(x - 8) = 0$

The roots are $x = 3$ and $x = 8$.

Answer: B. $3$ and $8$

6. Mrs. Santos asked how to solve $x^2 + 6x - 7 = 0$ by completing the square. What is the first step?

The process of completing the square involves making the left-hand side a perfect square trinomial:

1. Move the constant to the other side: $x^2 + 6x = 7$.
2. Take half of the coefficient of $x$, which is $\frac{6}{2} = 3$, and square it, which gives $9$.
3. Add $9$ to both sides of the equation.

The correct first step is to add the square of one-half of 6 to both sides.

Answer: B. Add the square of one-half of 6 on both sides of the equation.

---

Do you need more details or clarification on any of these steps?

Related Questions:

1. What is the discriminant of a quadratic equation, and how does it determine the nature of the roots?
2. How do you factor quadratic equations quickly?
3. What is the quadratic formula, and when should you use it?
4. What are the differences between completing the square and factoring?
5. How do you solve a quadratic equation using the method of graphing?

Tip: To check if a quadratic equation can be factored, look at the discriminant: $b^2 - 4ac$. If it's a perfect square, the quadratic is factorable.