To solve the equation t² – t – 12 = 0 by factoring, we first need to factor the quadratic expression on the left side.
We are looking for two numbers that multiply to -12 (the constant term) and add up to -1 (the coefficient of the linear term).
After examining the factors of -12, we find that -4 and 3 meet these criteria:
- -4 × 3 = -12
- -4 + 3 = -1
With these numbers, we can rewrite the middle term of the quadratic equation:
t² – 4t + 3t – 12 = 0
Next, we group the terms:
(t² – 4t) + (3t – 12) = 0
Now, we factor each group:
t(t – 4) + 3(t – 4) = 0
Now we can factor out the common factor, which is (t – 4):
(t – 4)(t + 3) = 0
To find the solutions, we can set each factor to zero:
- t – 4 = 0 ⟹ t = 4
- t + 3 = 0 ⟹ t = -3
Thus, the solutions to the equation t² – t – 12 = 0 are t = 4 and t = -3.