To find the completely factored form of the quadratic expression 9x² – 24x + 16, we can start by searching for two numbers that multiply to 9 times 16 = 144 (the product of the coefficient of x² and the constant term) and add up to -24 (the coefficient of x).
The two numbers that meet these criteria are -12 and -12, since:
- -12 + -12 = -24
- -12 times -12 = 144
Now, we can rewrite the middle term of the quadratic expression:
9x² – 12x – 12x + 16
Next, we group the terms:
(9x² – 12x) + (-12x + 16)
Now, factor out the common factors in each group:
3x(3x – 4) – 4(3x – 4)
Notice that we have a common binomial factor of (3x – 4):
(3x – 4)(3x – 4)
We can express this factorization as:
(3x – 4)²
So, the completely factored form of 9x² – 24x + 16 is:
(3x – 4)²