Sure, let's find the value of the coefficient [tex]\( B \)[/tex] in the factored form of the trinomial.
Given the trinomial:
[tex]\[ 2x^2 + 4xy - 48y^2 \][/tex]
The factored form is proposed as:
[tex]\[ 2(x + By)(x - 4y) \][/tex]
We need to find the value of [tex]\( B \)[/tex]. Let's expand the factored form [tex]\( 2(x + By)(x - 4y) \)[/tex] and equate it to the original trinomial:
1. First, expand [tex]\( (x + By)(x - 4y) \)[/tex]:
[tex]\[ (x + By)(x - 4y) = x^2 - 4xy + Bxy - 4By^2 \][/tex]
2. Now multiply the entire expression by 2:
[tex]\[ 2(x^2 - 4xy + Bxy - 4By^2) = 2x^2 - 8xy + 2Bxy - 8By^2 \][/tex]
3. This gives us:
[tex]\[ 2x^2 + (2B - 8)xy - 8By^2 \][/tex]
4. Now, equate this to the original trinomial to identify [tex]\( B \)[/tex]:
[tex]\[ 2x^2 + (2B - 8)xy - 8By^2 = 2x^2 + 4xy - 48y^2 \][/tex]
By comparing coefficients, we get:
[tex]\[ 2B - 8 = 4 \quad \text{(coefficient of the } xy \text{ term)} \][/tex]
[tex]\[ -8B = -48 \quad \text{(coefficient of the } y^2 \text{ term)} \][/tex]
5. Solve these equations one at a time:
From [tex]\( -8B = -48 \)[/tex]:
[tex]\[ B = 6 \][/tex]
Let’s verify if [tex]\( B = 6 \)[/tex] satisfies the first equation:
[tex]\[ 2B - 8 = 4 \][/tex]
Plugging [tex]\( B = 6 \)[/tex] in:
[tex]\[ 2(6) - 8 = 4 \][/tex]
[tex]\[ 12 - 8 = 4 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Both conditions are satisfied, confirming that:
[tex]\[ B = 6 \][/tex]
Therefore, the value of [tex]\( B \)[/tex] in the factored form is:
[tex]\[ \boxed{6} \][/tex]
