Equate quadratic equations that are equal to 100, as much as possible. Solutions should be a positive whole number.

To find quadratic equations equal to 100, such that their solutions are positive whole numbers, we begin by setting up the general quadratic equation in the form:

\[
ax^2 + bx + c = 100
\]

Here, we want the solutions for \(x\) to be positive integers. We will try different coefficients \(a\), \(b\), and \(c\) to identify appropriate quadratic equations. 

### Case 1: Simple Quadratic \(x^2 + bx + c = 100\)

1. **Quadratic Form: \(x^2 + bx - (100 - c) = 0\)**

   The discriminant for this quadratic equation is given by:

   \[
   \Delta = b^2 + 4(100 - c)
   \]

   For \(x\) to be a whole number, \(\Delta\) must be a perfect square.

   Let's try specific values:

   - \(x^2 - 20x + 100 = 0\): This does not have integer solutions.
   - \(x^2 - 10x + 24 = 0\): Discriminant \( \Delta = 10^2 - 4(1)(24) = 100 - 96 = 4 \), so \(x = \frac{10 \pm \sqrt{4}}{2} = 7 \text{ or } 3\).

     **Quadratic Equation:** \(x^2 - 10x + 24 = 100\)

     **Solutions:** \(x = 7\) or \(x = 3\) (both are positive integers).

### Case 2: Quadratic with Larger Leading Coefficient \(ax^2 + bx + c = 100\)

   - Try \(a = 1\), \(b = -15\), \(c = 56\):

     \[
     x^2 - 15x + 56 = 0
     \]
     The discriminant is \(15^2 - 4(1)(56) = 225 - 224 = 1\). Thus, \(x = \frac{15 \pm \sqrt{1}}{2} = 8 \text{ or } 7\).

     **Quadratic Equation:** \(x^2 - 15x + 56 = 100\)

     **Solutions:** \(x = 8\) or \(x = 7\) (both are positive integers).

### Summary:

Two possible quadratic equations where solutions are positive whole numbers, equating to 100, are:

1. \(x^2 - 10x + 24 = 100\), with solutions \(x = 7\) or \(x = 3\).
2. \(x^2 - 15x + 56 = 100\), with solutions \(x = 8\) or \(x = 7\).

Would you like more detailed explanations, or do you have any questions?

Here are 5 related questions to consider:

1. How can we determine if a quadratic equation will have integer solutions?
2. What are the key properties of the discriminant in quadratic equations?
3. Can we generate more quadratic equations that equate to other numbers?
4. What are the geometric interpretations of these quadratic equations?
5. How does the value of the constant term \(c\) affect the solutions of the quadratic equation?

**Tip:** When dealing with quadratic equations, always check the discriminant to determine if the roots are real and rational.

Explore different quadratic equations that equate to 100 and have positive whole number solutions. Learn about key algebraic concepts like the discriminant and the quadratic formula, suitable for students in Grades 9-12.

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