Problem 23 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}=144 $$

Step-by-Step Solution

Answer

x = ±12

Step 1 - Understand the Equation

The given equation is in the form of a perfect square: x^2 = 144

Step 2 - Apply the Square Root Property

Take the square root on both sides of the equation to solve for x. The square root property states that if a^2 = b, then a = ± sqrt(b). Applying this property: x = ± sqrt(144)

Step 3 - Simplify the Radicals

Calculate the square root of 144. sqrt(144) = 12 because 12 * 12 = 144. Therefore, x = ±12

Step 4 - Write the Final Solution

Combine the results to express the final solution: x = ±12.

Key Concepts

square root property

The Square Root Property is a very useful tool for solving quadratic equations, especially when the equation is in the form of a perfect square. The property states that if you have an equation where a square of a variable equals a constant, like \( x^2 = c \), you can find the variable by taking the square root of the constant.
This can be expressed as \( x = \pm \sqrt{c} \).
This means you need to consider both the positive and negative roots of the constant.
Let's apply this to our exercise:
For the equation \( x^2=144 \), take the square root of both sides to get \( x = \pm \sqrt{144} \).
This results in two possible solutions for x:

• \( x = +12 \)
• \( x = -12 \)

simplifying radicals

Simplifying radicals means breaking down a complex square root into its simplest form. This process makes it easier to understand and work with the values.
When simplifying a radical, you aim to find a simpler or more easily interpretable expression.
In our exercise, we looked at \( \sqrt{144} \).
The number 144 is a perfect square. It's the product of 12 multiplied by itself ( \( 12 \times 12 = 144 \) ).
Recognizing that allows us to replace \( \sqrt{144} \) with the simpler value of 12. Remember:

• If the number under the square root (the radicand) is a perfect square, your job is much easier.
• If it’s not a perfect square, you can often simplify it by factoring out the largest perfect square.

So, \( \sqrt{144} \) simplifies directly to 12, leading us to the final solution: \( x = \pm 12 \).

perfect square

A perfect square is a number that can be expressed as the product of an integer with itself. In other words, a perfect square is a square of a whole number.
Examples of perfect squares include 1 ( \( 1 \times 1 \) ), 4 ( \( 2 \times 2 \) ), 9 ( \( 3 \times 3 \) ), and so on.
In our exercise, the number 144 is a perfect square because it can be written as \( 12 \times 12 \).
Recognizing perfect squares helps in simplifying square roots, as seen when solving \( x^2 = 144 \).
Here’s a quick tip:

• It’s beneficial for you to memorize at least the perfect squares up to 15 squared (\( 15^2 = 225 \)) since they appear often in mathematical problems.

Mastering the identification of perfect squares will make working with quadratic equations and radicals a lot easier.