Certainly! To solve the expression [tex]\((7a - 4b + 5c)^2\)[/tex], we need to expand it using the square of a trinomial formula. The formula for the square of a trinomial [tex]\((x + y + z)^2\)[/tex] is:
[tex]\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz \][/tex]
For our specific case:
[tex]\[ (7a - 4b + 5c)^2 \][/tex]
We'll substitute [tex]\(x = 7a\)[/tex], [tex]\(y = -4b\)[/tex], and [tex]\(z = 5c\)[/tex] into the formula.
1. First, calculate the square of each term:
[tex]\[ x^2 = (7a)^2 = 49a^2 \][/tex]
[tex]\[ y^2 = (-4b)^2 = 16b^2 \][/tex]
[tex]\[ z^2 = (5c)^2 = 25c^2 \][/tex]
2. Next, calculate the pairwise products multiplied by 2:
[tex]\[ 2xy = 2 \cdot (7a) \cdot (-4b) = -56ab \][/tex]
[tex]\[ 2xz = 2 \cdot (7a) \cdot (5c) = 70ac \][/tex]
[tex]\[ 2yz = 2 \cdot (-4b) \cdot (5c) = -40bc \][/tex]
Now, combine all these terms together:
[tex]\[ (7a - 4b + 5c)^2 = 49a^2 + 16b^2 + 25c^2 - 56ab + 70ac - 40bc \][/tex]
Thus, the expanded form of [tex]\((7a - 4b + 5c)^2\)[/tex] is:
[tex]\[ 49a^2 + 16b^2 + 25c^2 - 56ab + 70ac - 40bc \][/tex]
