Solve for a : 6( a-1)^2 -9 (a-1) -42
6( a-1)^2 -9 (a-1) -42
2(a-1)^2 - 3(a-1) - 14 = 0
((a-1)+2)(2(a-1)-7) = 0
think you can take it from there?
6 (a^2 - 2a + 1) - 9a + 9 - 42 ... 6a^2 - 12a + 6 - 9a - 33 ... 6 a^2 - 21 a - 27
2 a^2 - 7 a - 9 ... (2a - 9) (a + 1)
To solve the equation for a, we can follow these steps:
Step 1: Distribute the 6 across the terms inside the parentheses:
6(a-1)² - 9(a-1) - 42
= 6(a² - 2a + 1) - 9(a-1) - 42
= 6a² - 12a + 6 - 9a + 9 - 42
Step 2: Combine like terms:
= 6a² - 21a - 27
Step 3: Set the equation equal to zero:
6a² - 21a - 27 = 0
Step 4: Simplify the equation if possible, by dividing through by common factors (if there are any). In this case, there are no common factors:
The equation remains: 6a² - 21a - 27 = 0
Step 5: Solve the quadratic equation. In this case, we can use factoring or quadratic formula.
Factoring:
To factorize, we need to find two numbers whose product is equal to the product of the coefficient of a² term and the constant term. In this case, the coefficient of a² is 6 and the constant term is -27. The product is -162.
We need to find two numbers whose product is -162 and whose sum is -21 (the coefficient of the a term).
After some trial and error, we can find that the two numbers are -27 and 6.
Therefore, we can rewrite the equation as:
(2a - 9)(3a + 3) = 0
Using the zero product property, we know that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero.
So, we have two possibilities:
(2a - 9) = 0 or (3a + 3) = 0
Solving each equation separately:
For (2a - 9) = 0:
2a - 9 = 0
Adding 9 to both sides:
2a = 9
Dividing by 2:
a = 9/2
For (3a + 3) = 0:
3a + 3 = 0
Subtracting 3 from both sides:
3a = -3
Dividing by 3:
a = -1
Therefore, the solutions to the equation 6(a-1)² - 9(a-1) - 42 = 0 are:
a = 9/2 or a = -1