Use inspection to solve the equation x2=1010

Question

Use inspection to solve the equation  x2=1010
 , selecting the correct solution set below. (1 point)
Responses

{−510,510}
left brace negative 5 superscript 10 baseline comma 5 superscript 10 baseline right brace

{105}

left brace 10 superscript 5 baseline right brace

{−55}
left brace negative 5 superscript 5 baseline right brace

{−105,105}
left brace negative 10 superscript 5 baseline comma 10 superscript 5 baseline right brace

GPT-4o mini · Answer

To solve the equation $ x^2 = 1010 $ by inspection, we need to find values of $ x $ such that when squared, they equal $ 1010 $. Taking the square root of both sides, we have: $$ x = \pm \sqrt{1010} $$ Now, we can estimate $ \sqrt{1010} $. We know that: $$ \sqrt{1000} \approx 31.62 $$ Since $ 1010 $ is slightly greater than $ 1000 $, we can deduce that $ \sqrt{1010} $ is slightly greater than $ 31.62 $. Comparing $ 32^2 $ and $ 31^2 $: $$ 31^2 = 961 \quad \text{and} \quad 32^2 = 1024 $$ Since $ 1010 $ is between $ 961 $ and $ 1024 $, we find that $ \sqrt{1010} $ is between $ 31 $ and $ 32 $. From this, we can deduce that the approximate values for $ x $ would be around $ \pm 31.62 $. Let's look for matching options: 1. $ \{ -510, 510 \} $ – Not correct as they are way off. 2. $ \{ 10^5 \} $ – This equals $ 100,000 $, not correct. 3. $ \{ -55 \} $ – Not correct since $ (-55)^2 = 3025 $. 4. $ \{ -10^5, 10^5 \} $ – Again, this is clearly off. None of these choices seem to suggest the correct approximate roots of $ \pm \sqrt{1010} $, which are roughly $ \pm 31.62 $. It appears that there might be confusion in expressing the options—likely they intended $ \{ -\sqrt{1010}, \sqrt{1010} \} $ in some form. But based on what is provided, there is no option that correctly represents $ \sqrt{1010} $ or $ -\sqrt{1010} $. If I had to select the response closest or more reasonable, I would suggest noting that none of the options is correct for solving $ x^2 = 1010 $.