Use factoring by grouping to solve the following equation.
4x^3 + 8x^2 − 9x − 18 = 0
x = 1 (smallest value)
x = 2
x = 3 (largest value)
4x^2(x+2)-9(x+2)=0
(x+2)(4x^2-9)=0
I assume you can take it from here. The second term is a difference of two squares, so it can be further factored.
To solve the equation 4x^3 + 8x^2 − 9x − 18 = 0 using factoring by grouping, follow these steps:
Step 1: Group the terms in pairs.
4x^3 + 8x^2 − 9x − 18 can be grouped as:
(4x^3 + 8x^2) − (9x + 18)
Step 2: Factor out the common terms from each group.
From the first group, factor out 4x^2:
4x^2(x + 2) − (9x + 18)
Step 3: Factor out the greatest common factor from the second group.
From the second group, factor out 9:
4x^2(x + 2) − 9(x + 2)
Step 4: Combine the common factor.
Combine the two terms with the common factor (x + 2):
(4x^2 − 9)(x + 2) = 0
Step 5: Set each factor equal to zero and solve for x.
Setting (4x^2 − 9) = 0:
4x^2 − 9 = 0
Solving this quadratic equation, we get two possible values for x:
x = √(9/4) (positive square root)
x = -√(9/4) (negative square root)
x = 3/2, -3/2
Setting (x + 2) = 0:
x + 2 = 0
Solving this linear equation, we get one possible value for x:
x = -2
Therefore, the solutions to the equation 4x^3 + 8x^2 − 9x − 18 = 0 are:
x = 3/2, -3/2, -2. The smallest value is -3/2, the largest value is 3/2.