Question

which number is closest to $-sqrt{212}$? -14.2 -14.5 -14.8 -13.9

Explanation:

Step1: Calculate the square of each option's absolute value

First, find the absolute value of each option: \(|-14.2| = 14.2\), \(|-14.5| = 14.5\), \(|-14.8| = 14.8\), \(|-13.9| = 13.9\). Then square them:

• For \(14.2\): \(14.2^2 = 14.2\times14.2 = 201.64\)
• For \(14.5\): \(14.5^2 = 14.5\times14.5 = 210.25\)
• For \(14.8\): \(14.8^2 = 14.8\times14.8 = 219.04\)
• For \(13.9\): \(13.9^2 = 13.9\times13.9 = 193.21\)

Step2: Compare the squared values with 212

Now, find the difference between each squared value and 212:

• For \(201.64\): \(|212 - 201.64| = 10.36\)
• For \(210.25\): \(|212 - 210.25| = 1.75\)
• For \(219.04\): \(|219.04 - 212| = 7.04\)
• For \(193.21\): \(|212 - 193.21| = 18.79\)

The smallest difference is \(1.75\), which corresponds to the number \(14.5\). Since we are looking for \(-\sqrt{212}\), the number closest is \(-14.5\) because the square of \(14.5\) is closest to \(212\) among the given options.

Answer:

-14.5