https://file.aiquickdraw.com/mathbot/file/1725288167964o2bcamh7.jpg

The equation given is:

\[
-4x^2 - 7x = -36
\]

To solve this equation for the positive value of \(x\), let's follow these steps:

### Step 1: Rearrange the equation
First, let's move all terms to one side of the equation to set it to zero:

\[
-4x^2 - 7x + 36 = 0
\]

### Step 2: Multiply the equation by -1
To make it easier to work with, multiply the entire equation by \(-1\) to eliminate the negative signs:

\[
4x^2 + 7x - 36 = 0
\]

### Step 3: Solve using the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(4x^2 + 7x - 36 = 0\), the coefficients are:
- \(a = 4\)
- \(b = 7\)
- \(c = -36\)

Now, plug these values into the quadratic formula:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4 \times 4 \times (-36)}}{2 \times 4}
\]

### Step 4: Calculate the discriminant
Calculate the discriminant:

\[
\text{Discriminant} = 7^2 - 4 \times 4 \times (-36) = 49 + 576 = 625
\]

Since the discriminant is 625, which is a perfect square, it simplifies to:

\[
\sqrt{625} = 25
\]

### Step 5: Solve for \(x\)
Now solve for \(x\) using the plus and minus values:

\[
x = \frac{-7 \pm 25}{8}
\]

This gives two possible solutions:

\[
x = \frac{-7 + 25}{8} = \frac{18}{8} = \frac{9}{4}
\]

\[
x = \frac{-7 - 25}{8} = \frac{-32}{8} = -4
\]

Since we are asked for the **positive solution**, the correct answer is:

\[
x = \frac{9}{4}
\]

### Final Answer:
The correct answer is **B) \(\frac{9}{4}\)**.

Would you like further details or have any questions? Here are some related questions:

1. How do you determine whether to use the quadratic formula for solving an equation?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you simplify square roots in the context of quadratic equations?
4. How can you factor quadratic equations instead of using the quadratic formula?
5. What are other methods to solve quadratic equations aside from factoring and using the quadratic formula?

**Tip:** Remember that the sign of the discriminant in a quadratic equation gives you valuable information about the nature of the roots (real and distinct, real and equal, or complex).

Learn how to solve the quadratic equation -4x^2 - 7x = -36 to find the positive value of x using the quadratic formula. This problem covers key algebraic concepts, including the quadratic formula and discriminant. Suitable for students in Grades 10-12.

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