To solve the given expression [tex]\(\frac{297 \times 297 - 203 \times 203}{94}\)[/tex] using the hint [tex]\(94 = 297 - 203\)[/tex], let’s follow a detailed, step-by-step approach:
1. Identify the components to be calculated:
- We need the square of 297 ([tex]\(297^2\)[/tex]).
- We need the square of 203 ([tex]\(203^2\)[/tex]).
2. Calculate the squares:
- [tex]\(297 \times 297 = 297^2\)[/tex]
- [tex]\(203 \times 203 = 203^2\)[/tex]
3. Subtract the second square from the first square:
- [tex]\(297^2 - 203^2\)[/tex]
4. Recognize the difference of squares formula:
- Before performing any direct multiplication and subtraction, notice that [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex] where [tex]\(a = 297\)[/tex] and [tex]\(b = 203\)[/tex].
5. Apply the difference of squares formula:
[tex]\[
297^2 - 203^2 = (297 + 203)(297 - 203)
\][/tex]
6. Substitute the values from the hint:
- [tex]\(a - b = 94\)[/tex] (given)
- [tex]\(a + b = 297 + 203 = 500\)[/tex]
7. Multiply the simplified terms:
[tex]\[
297^2 - 203^2 = 500 \times 94
\][/tex]
8. Calculate the product:
- [tex]\(500 \times 94 = 47000\)[/tex]
9. Put the calculated numerator into the original fraction:
[tex]\[
\frac{297^2 - 203^2}{94} = \frac{47000}{94}
\][/tex]
10. Perform the division:
[tex]\[
\frac{47000}{94} = 500
\][/tex]
11. Conclusion:
The numerator is 47000, and the final result is 500.
Thus, the detailed step-by-step solution shows that the expression [tex]\(\frac{297^2 - 203^2}{94}\)[/tex] simplifies to 500, with the calculated numerator being 47000.
