Use inspection to solve the equation x2=36169 , then select the correct solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Responses
{613}
left brace Start Fraction 6 over 13 End Fraction right brace
{−613,613}
left brace negative Start Fraction 6 over 13 End Fraction comma Start Fraction 6 over 13 End Fraction right brace
There are no real solutions.
There are no real solutions.
{−613}
left brace negative Start Fraction 6 over 13 End Fraction right brace
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page 19 of 19
To solve the equation \( x^2 = 36169 \) by inspection, we need to find the square root of 36169.
Inspecting the number 36169, we notice that it is a perfect square because the last two digits, 69, indicate that it could be the square of a number ending in 3 or 7 (since \(3^2 = 9\) and \(7^2 = 49\), which ends in 9 after carrying).
The first two digits, 361, are familiar because \(19^2 = 361\). This gives us a clue that the square root of 36169 might be a number close to 190 and ending in 3 or 7.
By further inspection or calculation, we can find that \( 189^2 = 35721 \) and \( 191^2 = 36481 \). This indicates that the number we are looking for is indeed between 189 and 191 and since the last digit of 36169 is 9, the square root's last digit must be 3.
Thus, \( 190^2 = 36100 \), and adding 69 to this gets us \( 36169 \) which means \( 190 + 3 = 193 \) or \( 190 - 3 = 187 \). However, 187 does not end in 3, so \( 193 \) is the correct square root.
So, \( x^2 = 36169 \) has two real solutions: \( x = 193 \) and \( x = -193 \).
The correct solution set is:
\{−193, 193\}