How do you factor 3x² + 10x - 8 = 0

To factor the quadratic equation 3x² + 10x - 8 = 0, follow these steps:

Identify a, b, and c: The quadratic equation is in the form ax² + bx + c = 0. In your case:

a = 3
b = 10
c = -8
Multiply a and c: Multiply the coefficients of x² (a) and the constant term (c).

a * c = 3 * -8 = -24
Find two numbers that multiply to give -24 and add to give 10 (the coefficient of x): The two numbers are 12 and -2, because:

12 * -2 = -24
12 + (-2) = 10
Rewrite the middle term using the two numbers found: Split the middle term (10x) into 12x and -2x: 3x² + 12x - 2x - 8 = 0

Factor by grouping: Group the terms in pairs: (3x² + 12x) - (2x + 8) = 0

Factor out the common factor from each group:

From the first group (3x² + 12x), factor out 3x: 3x(x + 4)
From the second group (-2x - 8), factor out -2: -2(x + 4)
Now the equation looks like this: 3x(x + 4) - 2(x + 4) = 0

Factor out the common binomial factor (x + 4): (x + 4)(3x - 2) = 0

Solve for x: Set each factor equal to 0 and solve for x:

x + 4 = 0 → x = -4
3x - 2 = 0 → 3x = 2 → x = 2/3
Thus, the solutions are:

x = -4
x = 2/3