Find the Missing Term in (x+?)(x-8) = x²-5x-24: Binomial Expansion

Complete the missing element

$(x+?)(x-8)=x^2-5x-24$

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

• Step 1: Expand the expression $(x+?)(x-8)$.
• Step 2: Compare the expanded expression to the given quadratic expression.
• Step 3: Determine the missing term by matching coefficients.

Now, let's work through each step:

Step 1: Write the left expression as $(x+b)(x-8) = x^2 - 8x + bx - 8b$.

Step 2: Compare the expanded form, $x^2 + (b-8)x - 8b$, to the given $x^2 - 5x - 24$.

Step 3: We see that the coefficients for $x$ should match, so we have $b-8 = -5$. Solving for $b$, we get:

• $b - 8 = -5 \rightarrow b = -5 + 8 \rightarrow b = 3$.

Optionally, verify by comparing the constant terms: $-8b = -24 \rightarrow b = 3$.

Therefore, the missing element in the binomial is $\boxed{3}$.