Compare (a-√20)² and a(a+20)+20: Algebraic Inequality Challenge

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

• Step 1: Simplify the expression $(a-\sqrt{20})^2$.
• Step 2: Simplify the expression $a(a+20) + 20$.
• Step 3: Compare the results of the two expressions.

Now, let's work through each step:

Step 1: We start with the expression $(a-\sqrt{20})^2$. Using the formula for the square of a difference, we get:

$(a-\sqrt{20})^2 = a^2 - 2a\sqrt{20} + 20$.

Step 2: Now we consider the expression $a(a+20) + 20$. Expanding this, we have:

$a(a+20) + 20 = a^2 + 20a + 20$.

Step 3: Now, we compare the two simplified expressions:

$a^2 - 2a\sqrt{20} + 20$ and $a^2 + 20a + 20$.

Both sides share an $a^2 + 20$, so we compare the remaining terms:

$-2a\sqrt{20}$ and $20a$.

Rewriting these as inequalities, since $0 < a$ and $-2\sqrt{20} \approx -8.944$, which is smaller than $20$. This gives:

$-2a\sqrt{20} < 20a$.

Thus, $(a-\sqrt{20})^2 < a(a+20) + 20$.

Therefore, the correct comparison sign is $<$.