Problem 12 - Elementary Algebra

Question

For the following problems, solve the equations. $$ c^{2}=36 $$

Step-by-Step Solution

Answer

Answer: The values of \(c\) that satisfy the equation are \(c=6\) and \(c=-6\).

Step 1: Evaluate the given equation

We are given the equation \(c^2=36\), which means that there is a number, \(c\), that when squared, is equal to 36. To solve for \(c\), we will use the square root properties.

Step 2: Apply the square root properties

To solve for \(c\), we will take the square root of both sides of the equation. The square root of a number squared is equal to the number itself, so that gives us: $$ \sqrt{c^2}=\sqrt{36} $$

Step 3: Calculate the square root

Calculate the square root of 36: $$ c=\pm\sqrt{36} $$ $$ c=\pm6 $$

Step 4: Write the final solution

The solution to the equation \(c^2=36\) is: $$ c=6 \textrm{ or } c=-6 $$

Key Concepts

Square Root Properties

Understanding the properties of square roots is crucial when solving quadratic equations. A square root, symbolized by \( \sqrt{} \), essentially asks the question 'what number multiplied by itself gives me this value?' For instance, if we ponder \( \sqrt{36} \), we're looking for a number that, when multiplied by itself, equals 36.

One of the fundamental properties of square roots is that \( \sqrt{x^2} = |x| \. Thus, when you square a number and then take the square root, you are left with the absolute value of the original number. This is why the equation \( c^2 = 36 \) gives us two answers: \( c = \pm6 \. The \pm sign indicates that both 6 and -6 are correct answers, as both of these numbers squared will return 36. It's this precise understanding of square roots that allows us to confidently solve equations like these.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. When faced with an equation like \( c^2 = 36 \), our objective is to find the value(s) of \( c \) that make the equation true. This process often involves isolating the variable on one side of the equation using algebraic operations such as addition, subtraction, multiplication, division, and taking roots.

Isolating the Variable

With the equation \( c^2 = 36 \) from our exercise, isolating \( c \) means undoing the squaring operation. Since an equation is like a balance scale, whatever we do to one side, we must also do to the other to keep it balanced. This balance is the core concept of solving an algebraic equation. After applying the square root to both sides, we simplify to find the possible solutions for \( c \. Always check your solutions by substituting them back into the original equation to ensure they work.

Radical Expressions

Expressions that contain a radical symbol—denoting roots, like square roots—are known as radical expressions. The key to handling these expressions is understanding how to simplify and manipulate them.

Simplifying Radical Expressions

Solving an equation that involves a radical requires that we simplify the radical if possible. In the context of our exercise, \( \sqrt{36} \) can be simplified because 36 is a perfect square. Perfect squares are numbers that are the square of integers, and their square roots are always integers as well. That's how \( \sqrt{36} \) simplifies directly to 6. However, not all radicals simplify so neatly, and we may encounter radicals that cannot be simplified to whole numbers. In those cases, we approximate their values or leave them in radical form, depending on the context of the problem.