How do you factor 2a^2 - 1 ?

To factor the expression 2a^2 - 1, follow these steps:

Step 1: Identify the type of quadratic expression
The given expression, 2a^2 - 1, is a quadratic expression. It is in the form of ax^2 + bx + c, where a = 2, b = 0, and c = -1.

Step 2: Determine the factors
We need to find two numbers that, when multiplied, give us the constant term (c = -1) and, when added or subtracted, give us the coefficient of the linear term (b = 0).

In this case, since b = 0, our two numbers will be 1 and -1 (the factors of -1).

Step 3: Rewrite the quadratic expression
Rewrite the x-term using the two numbers found in Step 2. Replace the coefficient of the linear term (b) with the sum of these numbers (1 - 1 = 0). Here's the updated expression:

2a^2 + 1a - 1a - 1

Step 4: Group the expression
Group the terms in pairs, considering the new coefficients obtained in Step 3:

(2a^2 + 1a) - (1a + 1)

Step 5: Factor by grouping
Factor each group separately:

a(2a + 1) - 1(1a + 1)

Step 6: Simplify further
Simplify the factorization expression by removing the common factors, if any:

(a - 1)(2a + 1)

Therefore, the expression 2a^2 - 1 factors into (a - 1)(2a + 1).