Solve by factoring: 3x^2 + 2x - 1 = 0
You know it's going to be
(3x+1)(x-1) or (3x-1)(x+1)
We want to end up with +2x, so it has to be
(3x-1)(x+1)
To solve the quadratic equation 3x^2 + 2x - 1 = 0 by factoring, we need to factorize the left side of the equation into two binomial expressions. Here's how to do it:
Step 1: Write down the equation: 3x^2 + 2x - 1 = 0
Step 2: Look at the coefficient of x^2, which is 3. You need to find two numbers that multiply to give you 3 times -1 (the constant term), and also add up to give you the coefficient of x, which is 2.
Step 3: List all possible factors of 3 and -1:
Factors of 3: 1, 3
Factors of -1: -1, 1
Step 4: Check all possible combinations to see if any of them add up to 2.
Possible combinations: (1, -1) and (-1, 1)
(1, -1): 1 + (-1) = 0 (Not equal to 2)
(-1, 1): -1 + 1 = 0 (Not equal to 2)
Since none of the combinations add up to 2, we cannot factorize the equation through simple factoring. In such cases, we can use the quadratic formula to find the solutions.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, where a = 3, b = 2, and c = -1, we can substitute these values into the quadratic formula to find the solutions:
x = (-2 ± √(2^2 - 4(3)(-1))) / (2(3))
x = (-2 ± √(4 + 12)) / 6
x = (-2 ± √16) / 6
x = (-2 ± 4) / 6
This simplifies to two solutions:
x = (-2 + 4) / 6 => x = 2 / 6 => x = 1/3
x = (-2 - 4) / 6 => x = -6 / 6 => x = -1
Therefore, the solutions to the quadratic equation 3x^2 + 2x - 1 = 0 are x = 1/3 and x = -1.