To determine which equation can be used to find the two numbers such that the sum of a number and its square is 42, let's start by defining the problem mathematically.
We are given that the sum of a number [tex]\( x \)[/tex] and its square [tex]\( x^2 \)[/tex] is 42. We can represent this relationship with the equation:
[tex]\[ x^2 + x = 42 \][/tex]
Now, let's analyze the given options to see which one matches this equation:
1. [tex]\( x^2 + x = 42 \)[/tex] - This equation directly matches our initial representation of the problem. It correctly states that the sum of [tex]\( x^) and \( x^2 \)[/tex] is 42.
2. [tex]\( x^2 + 2x = 42 \)[/tex] - This equation states that the sum of [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] is 42. This is not the same as the original problem since it includes [tex]\( 2x \)[/tex] instead of [tex]\( x \)[/tex].
3. [tex]\( x^2 + x + 42 = 0 \)[/tex] - This equation could be derived from our initial representation if we subtract 42 from both sides, leading to [tex]\( x^2 + x - 42 = 0 \)[/tex]. Although this equation is mathematically related to the original equation, it doesn't directly state the relationship given in the problem.
4. [tex]\( x^2 + 2x + 42 = 0 \)[/tex] - This equation states that the sum of [tex]\( x^2 \)[/tex], [tex]\( 2x \)[/tex], and 42 is zero. This equation does not represent the original problem at all, as it includes additional terms.
Based on this analysis, the correct equation that can be used to find the numbers for the given problem is:
[tex]\[ x^2 + x = 42 \][/tex]
