Solving the Math Equation: Finding the Missing Number

This solution explains how to find the value that replaces the question mark ('?') in the given mathematical equation. We will break down the problem step-by-step.

Understanding the Equation

The equation provided is:

$$ (4575 – 1740 – 2250) = ? – 39 + \sqrt{9025} $$

Our goal is to isolate the question mark ('?') and find its numerical value.

Step 1: Calculate the Left Side of the Equation

First, let's simplify the expression within the parentheses on the left side:

• Subtract the first two numbers: $4575 – 1740 = 2835$
• Subtract the next number from the result: $2835 – 2250 = 585$

So, the left side of the equation simplifies to 585.

Step 2: Calculate the Square Root on the Right Side

Next, we need to find the square root of 9025:

$$ \sqrt{9025} $$

To find the square root of 9025:

• We look for a number whose square is 9025. Since the number ends in 5, its square root must also end in 5.
• We can estimate: $90^2 = 8100$ and $100^2 = 10000$. The number is between 90 and 100.
• Let's try 95: $95 \times 95 = 9025$.

Alternatively, using the shortcut for squaring numbers ending in 5: $(N5)^2 = [N \times (N+1)] \text{ followed by } 25$. Here, N=9. So, $(9 \times (9+1)) \text{ followed by } 25 = (9 \times 10) \text{ followed by } 25 = 9025$.

Therefore, $\sqrt{9025} = 95$.

Step 3: Substitute the Calculated Values Back into the Equation

Now, substitute the simplified left side and the calculated square root back into the original equation:

$$ 585 = ? – 39 + 95 $$

Step 4: Simplify the Right Side of the Equation

Combine the known numbers on the right side:

$$ 585 = ? + (95 – 39) $$

$$ 585 = ? + 56 $$

Step 5: Solve for the Question Mark ('?')

To find the value of '?', subtract 56 from both sides of the equation:

$$ ? = 585 – 56 $$

$$ ? = 529 $$

Conclusion

By performing the arithmetic operations and simplifying the equation, we find that the value replacing the question mark is 529.