To factor the quadratic expression 8x² + 13x + 6, we follow a systematic approach:
- Identify the coefficients: In the expression ax² + bx + c, we have:
- a = 8
- b = 13
- c = 6
- Multiply a and c: We calculate:
- 8 (from a) * 6 (from c) = 48
- Find two numbers that multiply to 48 and add up to 13: We need two numbers whose product is 48 and sum is 13. Those numbers are:
- 3 and 16
- Rewrite the middle term: Replace 13x with 3x + 16x:
- 8x² + 3x + 16x + 6
- Group the terms: We will group the first two terms and the last two terms:
- (8x² + 3x) + (16x + 6)
- Factor out common factors: From each group, we can factor out:
- For the first group (8x² + 3x), factor out x: x(8x + 3)
- For the second group (16x + 6), factor out 2: 2(8x + 3)
- Combine the factors: Now we can combine our factored terms:
- x(8x + 3) + 2(8x + 3)
- Factor out the common binomial: Both terms have a common factor of (8x + 3):
- (8x + 3)(x + 2)
Therefore, the factorization of the quadratic expression 8x² + 13x + 6 is: