Question

Factor the perect square trinomial completely. 4w²-48w+144

Answer 1

To factor the perfect square trinomial 4w²-48w+144, we can rewrite it as (2w - 12)², confirming that it is indeed a perfect square trinomial.

Step-by-step explanation:

To factor the perfect square trinomial 4w²-48w+144 completely, we look for two identical binomials that multiply to give the original trinomial. A perfect square trinomial is of the form (aw - b)² = a²w² - 2abw + b². In this case, we have:

a² = 4 (since 4w² = (2w)²)
b² = 144 (since 144 is a perfect square and can be written as 12²)

Given that the middle term is -48w, we need to verify that -2ab equals -48. Here, -2 * 2w * 12 indeed equals -48w, satisfying the condition for a perfect square trinomial.

The factored form of the trinomial is:

(2w - 12)²

Therefore, the factored form of the equation 4w²-48w+144 is (2w - 12)².

Answer 2

The factored form of the perfect square trinomial (4w² - 48w + 144) is (2w - 12)².

Step-by-step explanation:

Given expression:

(4w² - 48w + 144)

Identify the perfect square trinomial pattern:

This expression can be recognized as a perfect square trinomial because it follows the pattern (a² - 2ab + b²), where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms.

Identify the square roots of the first and last terms:

The square root of (4w²) is (2w) because (2w)² = 4w²).

The square root of 144 is 12 because (12² = 144).

Apply the formula a² - 2ab + b² = (a - b)²:

Substitute the square roots identified earlier into the formula.

(a² - 2ab + b²) becomes (2w)² - 2 * 2w * 12 + (12)².

Simplify the expression:

(4w² - 48w + 144).

Factor the perfect square trinomial:

Using the formula (a² - 2ab + b² = (a - b)²):

4w² - 48w + 144 is factored as (2w - 12)²

So, by recognizing the pattern, finding the square roots of the first and last terms, and applying the formula for a perfect square trinomial, the expression (4w² - 48w + 144) is factored into (2w - 12)². This method helps quickly identify and factor perfect square trinomials by utilizing the pattern and square root relationships between the terms.