Rewrite the middle term as the sumof two terms and then factor completely.
Please show me the formula and how to use it correctly.
12w^2 + 19w + 4
19w can be written as 16w + 3w. Those are the cross product terms when multiplying 4w times 4 and 3w times 1. That is a clue that the polynomial can be factored as
(4w + 1)(3w + 4)
To rewrite the middle term as the sum of two terms and then factor completely, you can follow these steps:
1. Identify the coefficients of the quadratic equation. In this case, the coefficients are:
- Coefficient of the squared term: 12w^2
- Coefficient of the linear term: 19w
- Coefficient of the constant term: 4
2. To find the factors of the product of the coefficient of the squared term and the constant term (12w^2 * 4 = 48w^2), you need to determine two numbers whose product is 48w^2 and whose sum is equal to the coefficient of the linear term (19w).
In this example, we want to find two numbers that multiply to 48w^2 and add up to 19w.
3. Factorize the coefficient of the squared term multiplied by the constant term (48w^2), which can be written as a product of two binomials:
(aw + b)(cw + d)
4. Find the values of a, b, c, and d that satisfy the equation. In this case, we can rewrite the middle term as 16w + 3w, because 16w * 3w = 48w^2 and 16w + 3w = 19w.
5. Now, you can rewrite the original equation by breaking down the middle term into the sum of 16w + 3w:
12w^2 + 16w + 3w + 4
6. Group the terms and factor by grouping:
(12w^2 + 16w) + (3w + 4)
7. Factor out the greatest common factor (GCF) from each group:
4w(3w + 4) + 1(3w + 4)
8. Observe that both groups have a common factor of (3w + 4), so you can factor it out:
(4w + 1)(3w + 4)
Therefore, after rewriting the middle term as the sum of two terms and factoring completely, the equation 12w^2 + 19w + 4 becomes (4w + 1)(3w + 4).