To solve the problem, we need to find the value of a given polynomial [tex]\( x^2 + kx + 5 \)[/tex] at [tex]\( x = 5 \)[/tex], given that its value at [tex]\( x = 2 \)[/tex] is 15.
### Step 1: Express the given information
The polynomial is [tex]\( x^2 + kx + 5 \)[/tex]. We know that when [tex]\( x = 2 \)[/tex], the polynomial equals 15. Mathematically, this can be written as:
[tex]\[ 2^2 + 2k + 5 = 15 \][/tex]
### Step 2: Simplify and solve for [tex]\( k \)[/tex]
First, substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[ 4 + 2k + 5 = 15 \][/tex]
Combine like terms:
[tex]\[ 9 + 2k = 15 \][/tex]
Subtract 9 from both sides to isolate the term with [tex]\( k \)[/tex]:
[tex]\[ 2k = 6 \][/tex]
Divide both sides by 2 to solve for [tex]\( k \)[/tex]:
[tex]\[ k = 3 \][/tex]
Now we know that [tex]\( k \)[/tex] is 3.
### Step 3: Substitute [tex]\( k \)[/tex] back into the polynomial
The polynomial with [tex]\( k = 3 \)[/tex] is:
[tex]\[ x^2 + 3x + 5 \][/tex]
### Step 4: Find the value of the polynomial at [tex]\( x = 5 \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] into the polynomial:
[tex]\[ 5^2 + 3 \cdot 5 + 5 \][/tex]
Calculate each term:
[tex]\[ 25 + 15 + 5 \][/tex]
Combine the values:
[tex]\[ 45 \][/tex]
### Final Answer
The value of the polynomial [tex]\( x^2 + 3x + 5 \)[/tex] at [tex]\( x = 5 \)[/tex] is [tex]\( 45 \)[/tex].
