Verified Answer :

  • Identify the equation as a perfect square form: \((x - 14)^2 = 36\).
  • Apply the square root property by taking the square root of both sides: \(x - 14 = \pm \sqrt{36} = \pm 6\).
  • Solve the two linear cases: \(x - 14 = 6\) gives \(x = 20\), and \(x - 14 = -6\) gives \(x = 8\).
  • The solutions are 8 and 20, which matches option B: \(\boxed{B}\).

Explanation

  1. Understanding the Square Root Property

The given equation is \((x - 14)^2 = 36\). This is a quadratic equation that can be solved using the square root property. The square root property states that if an equation is in the form \(u^2 = c\), where \(c \geq 0\), its solutions are given by \(u = \pm\sqrt{c}\). In our case, \(u = x - 14\) and \(c = 36\).

  1. Applying the Square Root Property

To solve \((x - 14)^2 = 36\), we take the square root of both sides of the equation:
\[\sqrt{(x - 14)^2} = \pm\sqrt{36}\]
Since the square root of 36 is 6, this simplifies to:
\[x - 14 = \pm 6\]
This gives us two separate linear equations to solve:
1) \(x - 14 = 6\)
2) \(x - 14 = -6\)

  1. Solving the Resulting Equations and Finding the Answer

Now we solve each of the two linear equations:
For the first equation:
\[x - 14 = 6\]
\[x = 6 + 14\]
\[x = 20\]

For the second equation:
\[x - 14 = -6\]
\[x = -6 + 14\]
\[x = 8\]
The solutions to the equation are \(x = 8\) and \(x = 20\). Comparing these results with the given options, we find that they correspond to option B.

Examples

In fields like structural engineering, the square root property is frequently used to determine dimensions or limits when a squared relationship exists. For example, if you are designing a square support pillar and you know the area must be 36 square units to safely distribute a load, you would use this principle to find the side length (\(x-14\)). Understanding these algebraic steps helps engineers balance physical constraints with mathematical requirements to build structures safely.

Related Questions