How will I find the 2 numbers in the calculator we need to find two numbers that multiply to 225 and add to -34.

To solve the problem of finding two numbers that multiply to 225 and add to -34, let's break it down:

You are looking for two numbers, say \( x \) and \( y \), such that:
1. \( x \times y = 225 \)
2. \( x + y = -34 \)

### Step-by-Step Approach:

1. **Set up the equations:**
   \[
   x \times y = 225 \quad \text{(Multiplication condition)}
   \]
   \[
   x + y = -34 \quad \text{(Addition condition)}
   \]

2. **Substitute one equation into the other:**
   From the addition equation, solve for one variable in terms of the other. Let's solve for \( y \):
   \[
   y = -34 - x
   \]
   Now substitute this into the multiplication equation:
   \[
   x \times (-34 - x) = 225
   \]

3. **Simplify and solve the quadratic equation:**
   Distribute \( x \) on the left-hand side:
   \[
   -34x - x^2 = 225
   \]
   Rearrange the equation to get everything on one side:
   \[
   x^2 + 34x - 225 = 0
   \]
   This is a quadratic equation.

4. **Solve the quadratic equation:**
   You can solve it using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \( x^2 + 34x - 225 = 0 \), the coefficients are:
   - \( a = 1 \)
   - \( b = 34 \)
   - \( c = -225 \)

   Substitute these values into the quadratic formula:
   \[
   x = \frac{-34 \pm \sqrt{34^2 - 4(1)(-225)}}{2(1)}
   \]
   \[
   x = \frac{-34 \pm \sqrt{1156 + 900}}{2}
   \]
   \[
   x = \frac{-34 \pm \sqrt{2056}}{2}
   \]
   \[
   x = \frac{-34 \pm 45.33}{2}
   \]

5. **Find the two solutions:**
   Now, calculate the two possible values of \( x \):
   \[
   x_1 = \frac{-34 + 45.33}{2} = \frac{11.33}{2} = 5.665
   \]
   \[
   x_2 = \frac{-34 - 45.33}{2} = \frac{-79.33}{2} = -39.665
   \]

6. **Find the corresponding values of \( y \):**
   For \( x_1 = 5.665 \), \( y = -34 - 5.665 = -39.665 \).
   For \( x_2 = -39.665 \), \( y = -34 - (-39.665) = 5.665 \).

### Conclusion:
The two numbers are approximately 5.665 and -39.665.

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### Related Questions:
1. How do you apply the quadratic formula to solve real-world problems?
2. What are some shortcuts for factoring quadratic equations?
3. How do you verify the solution of a quadratic equation?
4. How can you simplify radicals in quadratic equations?
5. What are alternative methods to solve quadratic equations besides the quadratic formula?

**Tip:** When solving quadratic equations, always check if you can factor the equation before using the quadratic formula—it can save time!

How will I find the 2 numbers in the calculator we need to find two numbers that multiply to 225 and add to -34?

Discover how to solve the problem of finding two numbers that multiply to 225 and add to -34 using quadratic equations. This solution involves breaking down the problem into algebraic concepts and applying the quadratic formula, suitable for students in Grades 9-11.

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