how to solve 9x-9=-4x^2 and give me explation

To solve the quadratic equation \( 9x - 9 = -4x^2 \), we'll go through the following steps:

### Step 1: Rearrange the equation
We need to get everything on one side of the equation so that one side equals 0. This will turn the equation into standard quadratic form, which looks like \( ax^2 + bx + c = 0 \).

The given equation is:

\[
9x - 9 = -4x^2
\]

First, add \( 4x^2 \) to both sides to move the quadratic term to the left:

\[
4x^2 + 9x - 9 = 0
\]

### Step 2: Apply the quadratic formula
The quadratic formula is used to solve equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula is:

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

In this equation, \( a = 4 \), \( b = 9 \), and \( c = -9 \).

### Step 3: Calculate the discriminant
The discriminant \( \Delta \) is the part under the square root in the quadratic formula, \( \Delta = b^2 - 4ac \).

\[
\Delta = 9^2 - 4(4)(-9)
\]
\[
\Delta = 81 + 144 = 225
\]

Since the discriminant is positive, there will be two real solutions.

### Step 4: Solve for \( x \)
Now substitute the values of \( a \), \( b \), and \( \Delta \) into the quadratic formula:

\[
x = \frac{-9 \pm \sqrt{225}}{2(4)}
\]
\[
x = \frac{-9 \pm 15}{8}
\]

This gives two possible solutions:

1. \( x_1 = \frac{-9 + 15}{8} = \frac{6}{8} = \frac{3}{4} \)
2. \( x_2 = \frac{-9 - 15}{8} = \frac{-24}{8} = -3 \)

### Final Answer:
The solutions to the equation \( 9x - 9 = -4x^2 \) are:

\[
x = \frac{3}{4} \quad \text{or} \quad x = -3
\]

---

Would you like a more detailed explanation of any of the steps? Here are 5 related questions you might find helpful:

1. How does the discriminant determine the number of real solutions for a quadratic equation?
2. Can you explain how to factor a quadratic equation when it's factorable?
3. What are the implications if the discriminant is zero or negative?
4. How do you complete the square to solve a quadratic equation?
5. When should you use the quadratic formula over factoring?

**Tip**: Always check if a quadratic equation can be factored easily before using the quadratic formula. This can save time!

how to solve 9x-9=-4x^2 and give me explanation

Learn how to solve the quadratic equation 9x - 9 = -4x^2 with a detailed solution. This guide covers important algebraic concepts, including the quadratic formula and discriminant, for students in Grades 9-10.

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