The given expression is a quadratic polynomial written as 9x² – 24x + 16. To find the completely factored form, we will use the method of factoring by grouping or the quadratic formula, though in this case, factoring by grouping is more straightforward.
First, we identify the coefficients:
- A = 9 (the coefficient of x²)
- B = -24 (the coefficient of x)
- C = 16 (the constant term)
Next, we look for two numbers that multiply to A × C = 9 × 16 = 144 and add to B = -24. The numbers that fulfill these conditions are -12 and -12 since:
- -12 × -12 = 144
- -12 + -12 = -24
Now, we can rewrite the middle term of the polynomial:
9x² – 12x – 12x + 16
Next, we group the terms:
(9x² – 12x) + (-12x + 16)
Now, we factor each group:
- From the first group, we can factor out 3x: 3x(3x – 4)
- From the second group, we can factor out -4: -4(3x – 4)
So we rewrite the expression as:
3x(3x – 4) – 4(3x – 4)
Now we can factor out the common term, which is (3x – 4):
(3x – 4)(3x – 4) or (3x – 4)²
In conclusion, the completely factored form of the expression 9x² – 24x + 16 is: