Factor:
w2-12w+36
The Correct Answer and Explanation is :
Let’s factor the quadratic expression w2−12w+36w^2 – 12w + 36.
Step 1: Recognize the structure
The quadratic expression is in the standard form: ax2+bx+cax^2 + bx + c
In our case:
- a=1a = 1 (since the coefficient of w2w^2 is 1)
- b=−12b = -12
- c=36c = 36
Step 2: Factor the expression
We need to find two numbers that multiply to cc (36) and add up to bb (-12).
- The factors of 36 are: 1,36;2,18;3,12;4,9;6,61, 36; 2, 18; 3, 12; 4, 9; 6, 6
- We need to check which pair of factors of 36 adds up to -12. The pair −6-6 and −6-6 works because: −6+(−6)=−12and−6×−6=36-6 + (-6) = -12 \quad \text{and} \quad -6 \times -6 = 36
Step 3: Rewrite the quadratic as a perfect square
Given that both numbers are −6-6, we can factor the expression as: w2−12w+36=(w−6)(w−6)w^2 – 12w + 36 = (w – 6)(w – 6)
This is a perfect square trinomial, and it can be written as: (w−6)2(w – 6)^2
Final Answer:
w2−12w+36=(w−6)2w^2 – 12w + 36 = (w – 6)^2
Explanation:
The process involves identifying two numbers that multiply to the constant term (36) and add up to the coefficient of the linear term (-12). Once we find these numbers, we can rewrite the quadratic as the square of a binomial. In this case, the numbers are both −6-6, which simplifies the expression into a perfect square trinomial. Therefore, the factored form is (w−6)2(w – 6)^2, indicating that the original quadratic expression is the square of the binomial w−6w – 6. This method of factoring works efficiently for perfect square trinomials, where both the linear and constant terms are related by a consistent factor.