How can we use the quadratic formula to solve the equation 
x
2
+
100
x
−
2400
=
0
x 
2
 +100x−2400=0?

To solve the quadratic equation \( x^2 + 100x - 2400 = 0 \) using the quadratic formula, follow these steps:

### Step 1: Identify the coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 100 \) (coefficient of \( x \))
- \( c = -2400 \) (constant term)

### Step 2: Write down the quadratic formula
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 3: Substitute the values of \( a \), \( b \), and \( c \) into the formula
\[
x = \frac{-100 \pm \sqrt{100^2 - 4(1)(-2400)}}{2(1)}
\]

### Step 4: Simplify under the square root (the discriminant)
\[
x = \frac{-100 \pm \sqrt{10000 + 9600}}{2}
\]
\[
x = \frac{-100 \pm \sqrt{19600}}{2}
\]

### Step 5: Calculate the square root
\[
x = \frac{-100 \pm 140}{2}
\]

### Step 6: Solve for the two possible values of \( x \)
- **For the positive root**:
  \[
  x = \frac{-100 + 140}{2} = \frac{40}{2} = 20
  \]
- **For the negative root**:
  \[
  x = \frac{-100 - 140}{2} = \frac{-240}{2} = -120
  \]

### Final Solutions:
The solutions to the equation \( x^2 + 100x - 2400 = 0 \) are:
\[
x = 20 \quad \text{or} \quad x = -120
\]

Would you like further details on these steps or explanations on related concepts? Here are five related questions you might consider:

1. Why do we consider both the positive and negative roots in the quadratic formula?
2. What is the significance of the discriminant in determining the nature of the roots?
3. How would the solutions change if the quadratic equation had a different constant term?
4. Can the quadratic formula be used for equations where \( a \) is not equal to 1?
5. How do the roots relate to the graph of the quadratic function \( y = x^2 + 100x - 2400 \)?

**Tip:** Always simplify the discriminant as much as possible before taking the square root to make the calculations easier.

Learn how to solve the quadratic equation x^2 + 100x - 2400 using the quadratic formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and solving quadratic equations. Suitable for students in Grades 9-12.

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How can we use the quadratic formula to solve the equation x 2 + 100 x − 2400

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