Verified Answer :

  • Evaluate the square root: \(\sqrt{16} = 4\).
  • Calculate the two values: \(5 + 4 = 9\) and \(5 - 4 = 1\).
  • The correct values are \(\boxed{9, 1}\).

Explanation

  1. Analyzing the expression

The given expression is \(x = 5 \pm \sqrt{16}\). This notation implies two separate values for \(x\): one resulting from adding the square root of 16 to 5, and the other resulting from subtracting the square root of 16 from 5. First, we identify the value of the square root term: \(\sqrt{16} = 4\).

  1. Calculating the values of x

Now that we have the square root term, we can substitute it back into the original expression. The two cases for \(x\) are:

  1. \(x = 5 + 4\)
  2. \(x = 5 - 4\)

Calculating these:

  1. \(x = 9\)
  2. \(x = 1\)

Comparing these results with the given options \(-11, 21, 9, 1\), we identify the two correct values as 9 and 1.

Examples

This type of calculation is useful when solving quadratic equations of the form \((x-h)^2 = k\), which is a common technique for finding the roots of parabolas in geometry or physics. For example, if you are calculating the time a projectile stays above a certain height, you might end up with an expression like \(t = t_0 \pm \Delta t\), where \(t_0\) is the midpoint time and \(\Delta t\) represents the duration until the object hits the ground, allowing you to find the exact launch and landing times.

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