Problem 53 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (x-8)^{2}=27 $$

Step-by-Step Solution

Answer

x = 8 + 3\sqrt{3} or x = 8 - 3\sqrt{3}

Step 1 - Isolate the squared term

The equation is already given in a form where the squared term \( (x - 8)^{2}= 27 \) is isolated. So, no further isolation is required.

Step 2 - Apply the Square Root Property

Take the square root of both sides of the equation. Remember to consider both positive and negative roots. So, taking the square root we get: \[ \sqrt{(x - 8)^2} = \pm \sqrt{27} \].

Step 3 - Simplify the Radicals

Simplify \( \sqrt{27} \). Since 27 can be factored into 9 and 3, we get \[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \]. Thus, \[ \sqrt{(x - 8)^2} = \pm 3\sqrt{3} \]

Step 4 - Solve for x

Now solve for x: \[ x - 8 = \pm 3\sqrt{3} \]. This results in two equations: \[ x - 8 = 3\sqrt{3} \] and \[ x - 8 = -3\sqrt{3} \].

Step 5 - Isolate x

Isolate x in both equations: \[ x = 8 + 3\sqrt{3} \] and \[ x = 8 - 3\sqrt{3} \]. These are the solutions to the original equation.

Key Concepts

Square Root Property

When solving quadratic equations where the variable is squared, we often use the square root property. This property states that if \((x-a)^2 = b\), then \((x-a) = \pm \sqrt{b}\).
This method is useful because it allows us to handle square terms directly by taking the square root of both sides.
However, always remember both the positive and negative roots.
In our exercise, we first isolated the squared term \((x - 8)^2\) and then applied the square root property:
\( \sqrt{(x - 8)^2} = \pm \sqrt{27} \).
This step gives us two potential solutions involving radicals, which we need to simplify.

Simplifying Radicals

Simplifying radicals ensures our answers are in their simplest form.
In the exercise, we needed to simplify \(\sqrt{27}\).
We did that by finding factors of 27 that are perfect squares:
\(\sqrt{27} = \sqrt{9 \times 3} \).
Since \(\sqrt{9}\) is 3, this simplifies to 3\sqrt{3}.
Thus, \(\sqrt{27}\) becomes \(\3\sqrt{3}\).
Simplifying radicals makes it easier to work with the solutions.
Now that we have \[ \sqrt{(x-8)^2} = \pm 3\sqrt{3} \], we proceed to solve for x.

Solving for Variable

Solving for a variable means isolating it on one side of the equation.
After applying the square root property and simplifying the radicals, we get \[ x-8 = \pm 3\sqrt{3} \].
This translates into two equations:
\((x-8 = 3\sqrt{3}) \) and \((x-8 = -3\sqrt{3}) \).
To isolate x in each equation, we add 8 to both sides:
\[ x = 8 + 3\sqrt{3} \] and \[ x = 8 - 3\sqrt{3} \].
These are the solutions to our equation and show how the value of x can be expressed in terms of radicals.
Always double-check these steps to make sure you didn't miss anything.