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Chapter 9: Problem 53

Solve using the Square Root Property. $$a^{2}-18=0$$

Short Answer

Expert verified
a = \( \pm 3\sqrt{2} \)

Step by step solution

01

Isolate the squared term

To solve the equation using the Square Root Property, start by isolating the squared term. The given equation is \[a^{2} - 18 = 0\] Add 18 to both sides of the equation to isolate the squared term: \[a^2 = 18\]
02

Apply the Square Root Property

Use the Square Root Property, which states that if \[x^2 = k\], then \[x = \pm \sqrt{k}\]. Apply this property to the isolated squared term: \[a = \pm \sqrt{18}\]
03

Simplify the square root

Simplify the square root expression: \[\sqrt{18}\] Rewrite \[18\] as \[9 \times 2\], then use the property of square roots: \[a = \pm \sqrt{9 \times 2} = \pm \sqrt{9} \times \sqrt{2} = \pm 3\sqrt{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the squared term
To solve equations using the Square Root Property, the first step is to isolate the squared term.
The given equation is \[a^{2} - 18 = 0\].
Start by moving constant terms to the other side. This can be done by adding or subtracting terms.
In this example, add \18\ to both sides:
\[a^2 - 18 + 18 = 18\]
Simplified, it becomes:
\[a^2 = 18\]
Now your squared term, \[a^2\], is isolated. It鈥檚 ready for the next step.
Making sure the squared term is alone simplifies further operations.
Solving quadratic equations
After isolating the squared term, the next step is to solve the equation using the Square Root Property. The property states that if \[x^2 = k\], then \[x = \pm \sqrt{k}\].
Using this, apply the property to \[a^2 = 18\].
This results in:
\[a = \pm \sqrt{18}\]
This says the solution to the equation could be either of the positive or negative square root of \18\. Always remember the \pm \ symbol indicates two solutions.
This step is crucial in finding the general solution.
Simplifying square roots
Simplifying square roots makes the final solution cleaner and easier to understand. In our case, we have \[a = \pm \sqrt{18}\].
Break down the square root by finding factors that are perfect squares:
\[18 = 9 \times 2\].
Apply the square root to each factor separately:
\[\begin{split} a &= \pm \sqrt{18} \ = \pm \sqrt{9 \times 2} \ = \pm \sqrt{9} \times \sqrt{2} \ = \pm 3 \sqrt{2} \end{split}\]
Now you have a simplified form \[a = \pm 3 \sqrt{2}\]
Simplifying ensures the final answer is easy to interpret.

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