Question
Factor completely: \( 9 x^{2}-100=\square \)
Ask by Stanley Crawford. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(9x^{2} - 100 = (3x - 10)(3x + 10)\)
Solution
To factor the expression \( 9x^{2} - 100 \) completely, we can recognize it as a **difference of squares**. The difference of squares formula is:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Let's apply this to the given expression:
1. **Identify the squares:**
- \( 9x^{2} = (3x)^2 \)
- \( 100 = 10^2 \)
2. **Apply the difference of squares formula:**
\[
9x^{2} - 100 = (3x)^2 - 10^2 = (3x - 10)(3x + 10)
\]
So, the completely factored form is:
\[
(3x - 10)(3x + 10)
\]
**Answer:**
\((3x - 10)\, (3x + 10)\)
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Simplify this solution The Deep Dive
The expression \(9 x^{2}-100\) is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a + b)(a - b)\). Here, \(9x^2\) is \((3x)^2\) and \(100\) is \(10^2\). Thus, we can write: \[ 9 x^{2}-100 = (3x)^2 - (10)^2 = (3x + 10)(3x - 10) \] Now you've got the expression factored completely! To check your work with a quick mental exercise, you can expand \((3x + 10)(3x - 10)\) back out to see if you return to the original form. This principle of factoring is particularly useful in solving quadratic equations or simplifying expressions, so keep practicing this technique!
