How many roots are there of the following equation? Find them.
32x+2 − 3x+3 − 3x + 3 = 0
32x+2 − 3x+3 – 3x + 3 = 0. Let us try grouping:
32x+2 − 3x+3 − 3x + 3 = 0
3x · 3x + 2 − 31 · 3x+2 − (3x − 3) = 0
(3x − 3)(3x+2) − 1(3x − 3) = 0
3x · 3x + 2 − 31 · 3x+2 − (3x − 3) = 0
(3x − 3)(3x+2) − 1(3x − 3) = 0
So we’ve factored it: (3x+2 − 1)(3x − 3) = 0
→ (9·3x − 1)(3x − 3) = 0
9·3x − 1 = 0 → 3x = 1/9 → x = -2
3x − 3 = 0 → 3x = 3 → x = 1. We’ve got two roots.
3x − 3 = 0 → 3x = 3 → x = 1. We’ve got two roots.
Check:
x = -2 : 3-4+2 − 3-2+3 − 3-2 + 3 = 0 ✓
x = 1 : 34 − 34 − 31 + 3 = 0 ✓
x = 1 : 34 − 34 − 31 + 3 = 0 ✓