Final answer:
To solve the quadratic equation 3a^2 - 13a - 10 = 0 by factoring, we find the numbers -15 and +2 which help us to rewrite the equation as (3a + 2)(a - 5) = 0. Setting each factor equal to zero gives us the solutions a = -2/3 and a = 5.
Step-by-step explanation:
To solve the quadratic equation 3a2 - 13a - 10 = 0 by factoring, we look for two numbers that multiply to give the product of the coefficient of a2 (which is 3) and the constant term (which is -10), and at the same time, these two numbers must add up to the coefficient of the 'a' term (which is -13).
These two numbers are -15 and +2 because (-15) * (+2) = -30 (which is 3 * -10) and (-15) + (+2) = -13. Therefore, we can write the equation as (3a + 2)(a - 5) = 0.
By applying the zero product property, we set each factor equal to zero and solve for 'a':
- 3a + 2 = 0 → a = -2/3
- a - 5 = 0 → a = 5
Thus, the solutions for the equation are a = -2/3 and a = 5.
We can check the answers by plugging them back into the original equation and confirming that they satisfy it.