Problem 61 - Elementary Algebra

Question

\(a^{2}=75\)

Step-by-Step Solution

Answer

\(a = \pm 5 \sqrt{3}\)

Step 1: Understand the Equation

Identify that this is a quadratic equation where a variable is squared. The given equation is \(a^{2} = 75\).

Step 2: Isolate the Variable

To solve for \(a\), take the square root of both sides of the equation. This gives \(a = \pm \sqrt{75}\).

Step 3: Simplify the Square Root

Simplify \(\sqrt{75}\) by expressing it as a product of prime factors: \(75 = 25 \times 3 = 5^{2} \times 3\). Therefore, \(\sqrt{75} = \sqrt{5^{2} \times 3} = 5 \sqrt{3}\).

Step 4: Final Answer

Combine the results to write the final answer. Thus, \(a = \pm 5 \sqrt{3}\).

Key Concepts

Square Root Property

The Square Root Property is very useful for solving quadratic equations. When you see an equation where a variable is squared, like \(a^{2}=75\), you can use this property.
The property says that if \(x^{2}=k\), then \(x = \pm \sqrt{k}\).
In our example, since \(a^{2}=75\), we apply the Square Root Property to get \(a = \pm \sqrt{75}\).
The \(\pm\) symbol indicates that we need to consider both the positive and negative solutions.
So, for quadratic equations, always remember the Square Root Property to quickly isolate your variable.

Prime Factorization

Prime Factorization helps us break down a number into its basic building blocks.
For instance, to simplify \(\root 75 \), we use prime factorization.
Here's how it works:
1. Start with the number 75.
2. Recognize that 75 can be written as a multiplication of smaller numbers: \( 75 = 25 \times 3 \).
3. Next, break 25 down further into prime factors: \( 25 = 5^{2} \).
So, \( 75 = 5^{2} \times 3 \).
By expressing 75 as \ 5^{2} \times 3 \, we prepare it for further simplification.

Simplifying Radicals

Simplifying radicals involves reducing square roots to their simplest form.
Using our prime factorized result from above, \( \sqrt{75} = \sqrt{5^{2} \times 3} \).
The \( 5^{2} \) inside the square root can be taken out as 5, making the expression much simpler:
\( \sqrt{75} = 5 \sqrt{3} \).
This process makes problems easier to handle and solutions more neat.
So for any radical, find its prime factors, take out pairs of primes, and simplify!