Question

the expression $n^2 + n$ represents the total number of blocks used in the $n$th figure of a pattern.
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use the equation $n^2 + n = 56$ to determine which figure has 56 blocks.
figure \boxed{} has 56 blocks.

Explanation:

Step1: Rearrange the equation

We start with the equation \( n^2 + n = 56 \). Rearrange it to the standard quadratic form \( n^2 + n - 56 = 0 \).

Step2: Factor the quadratic equation

We need to find two numbers that multiply to -56 and add up to 1. The numbers are 8 and -7. So, the factored form is \( (n + 8)(n - 7) = 0 \).

Step3: Solve for n

Setting each factor equal to zero gives \( n + 8 = 0 \) or \( n - 7 = 0 \). Solving these, we get \( n = -8 \) or \( n = 7 \). Since \( n \) represents the figure number, it must be positive. So, we discard \( n = -8 \).

Answer:

7