Solve by factoring:
10t^2 – t − 2 = 0
try (5t +2)
To solve the given quadratic equation 10t^2 - t - 2 = 0 by factoring, we need to factorize the quadratic expression on the left side of the equation.
First, we multiply the coefficient of t^2 (10) with the constant term (-2), which gives us -20. We need to find two numbers that multiply to give -20 and add up to the coefficient of t (-1).
The pairs of numbers that multiply to -20 are:
1 and -20
2 and -10
4 and -5
Among these pairs, the pair that adds up to -1 is 2 and -10.
Now, we rewrite the middle term (-t) using these two numbers:
- t = 2t - 10t
This allows us to rewrite the given quadratic equation as:
10t^2 + 2t - 10t - 2 = 0
Now, we can group the terms and factor by grouping:
(10t^2 + 2t) - (10t + 2) = 0
Taking out the common factors from each group:
2t(5t + 1) - 2(5t + 1) = 0
We can see that we have a common binomial factor of (5t + 1), so we can factor it out:
(2t - 2)(5t + 1) = 0
Now, we have two factors that multiply to give zero. According to the zero product property, if a product of two factors is equal to zero, then at least one of the factors must be equal to zero.
So, we set each factor equal to zero and solve for t:
2t - 2 = 0
2t = 2
t = 1
and
5t + 1 = 0
5t = -1
t = -1/5
Therefore, the solutions to the given quadratic equation 10t^2 - t - 2 = 0 are t = 1 and t = -1/5.