To find the value of [tex]\( k \)[/tex] such that the coefficient of [tex]\( x^2 \)[/tex] is equal to the coefficient of [tex]\( x \)[/tex] in the expansion of the expression [tex]\((x-3)(x^2 + 5x + k)\)[/tex], follow these steps:
1. Expand the expression [tex]\((x-3)(x^2 + 5x + k)\)[/tex]:
Let's distribute [tex]\((x-3)\)[/tex] to each term in [tex]\((x^2 + 5x + k)\)[/tex]:
[tex]\[ (x-3)(x^2 + 5x + k) = x(x^2 + 5x + k) - 3(x^2 + 5x + k) \][/tex]
Distributing [tex]\( x \)[/tex] and [tex]\(-3\)[/tex] to each term inside the parentheses:
[tex]\[ = x^3 + 5x^2 + kx - 3x^2 - 15x - 3k \][/tex]
2. Combine like terms:
Now, combine the terms with the same powers of [tex]\( x \)[/tex]:
[tex]\[ = x^3 + (5x^2 - 3x^2) + (kx - 15x) - 3k \][/tex]
[tex]\[ = x^3 + 2x^2 + (k - 15)x - 3k \][/tex]
3. Identify the coefficients:
From the expanded and simplified form [tex]\( x^3 + 2x^2 + (k - 15)x - 3k \)[/tex], we can identify:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 2 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( k - 15 \)[/tex].
4. Set up the equation:
According to the problem, the coefficient of [tex]\( x^2 \)[/tex] should be equal to the coefficient of [tex]\( x \)[/tex]:
[tex]\[ 2 = k - 15 \][/tex]
5. Solve for [tex]\( k \)[/tex]:
To find [tex]\( k \)[/tex], solve the equation [tex]\( 2 = k - 15 \)[/tex]:
[tex]\[ 2 = k - 15 \][/tex]
Add 15 to both sides of the equation:
[tex]\[ 2 + 15 = k \][/tex]
[tex]\[ k = 17 \][/tex]
So, the value of [tex]\( k \)[/tex] is 17.
