Complete the Square: Transform x²+9x+24 into (x+5)²

$y=x^2+9x+24$

Which expression should be added to y so that:

$y=(x+5)^2$

To solve this problem, we'll follow these steps:

• Step 1: Expand $(x+5)^2$ using the formula for the square of a sum.
• Step 2: Compare the expanded expression with $x^2 + 9x + 24$.
• Step 3: Determine what additional expression is needed to make the two expressions equal.

Let's go through these steps in detail:
Step 1: First, we expand $(x+5)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$:
$(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25$.

Step 2: Now, compare the expanded expression $x^2 + 10x + 25$ with the given $x^2 + 9x + 24$.
We notice that the linear term 10x needs to be replaced or adjusted with 9x, and the constant 25 with 24.

Step 3: To make the expressions equal, find the difference in linear and constant terms:
$10x + 25$ must equal $9x + 24 + k$ (where $k$ is what we need to add):
Equating them, we get $10x + 25 = 9x + 24 + k$.
Solve for $k$:
$10x + 25 - 9x - 24 = k$.
$x + 1 = k$.

Therefore, the expression that should be added is $x+1$.