Sure, let's break down the given expression [tex]\( 196q^2 - 1 \)[/tex] step-by-step.
1. Identify the constants and variables:
- The coefficient of [tex]\( q^2 \)[/tex] is 196, which is a constant.
- [tex]\( q \)[/tex] is the variable.
- The entire term [tex]\( 196q^2 \)[/tex] is a quadratic term (highest power of [tex]\( q \)[/tex] is 2).
- The constant term is -1.
2. Check for special forms:
- Notice that [tex]\( 196q^2 - 1 \)[/tex] resembles the difference of squares formula. The difference of squares formula states that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
3. Recognize the squares:
- Here, [tex]\( 196q^2 \)[/tex] can be written as [tex]\( (14q)^2 \)[/tex] because [tex]\( 196 = 14^2 \)[/tex].
- Similarly, the number 1 is [tex]\( 1^2 \)[/tex].
4. Apply the difference of squares formula:
- If we set [tex]\( a = 14q \)[/tex] and [tex]\( b = 1 \)[/tex], then [tex]\( 196q^2 - 1 \)[/tex] becomes [tex]\( (14q)^2 - 1^2 \)[/tex].
- Applying the difference of squares formula [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex], we get:
[tex]\[
(14q)^2 - 1^2 = (14q - 1)(14q + 1)
\][/tex]
5. Factored form:
- Thus, the expression can be factored into:
[tex]\[
196q^2 - 1 = (14q - 1)(14q + 1)
\][/tex]
In conclusion, the expression [tex]\( 196q^2 - 1 \)[/tex] can be factored as:
[tex]\[
196q^2 - 1 = (14q - 1)(14q + 1)
\][/tex]
