Problem 25 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=121 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 11\) and \(x = -11\)

Step 1: Isolate the square term

In this equation \(x^{2} = 121\), the square term \(x^{2}\) is already isolated

Step 2: Take the square root of both sides

Taking the square root of both sides, remember that it can yield both positive and negative results. As such, we reach: \(\sqrt{x^{2}} = \pm \sqrt{121}\)

Step 3: Simplify the Roots

Simplify both square roots to obtain: \(x = \pm 11\)

Key Concepts

Square Roots

Understanding the concept of square roots is fundamental when solving quadratic equations, such as \(x^2 = 121\). A square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 121 is 11, since \(11 \times 11 = 121\).

When we apply this concept to our equation, taking the square root of both sides is necessary in order to 'undo' the squaring of \(x\). It's important to note that every positive number has two square roots: a positive and a negative root, represented by \(\pm\). This is because \((-11) \times (-11)\) also equals 121. Therefore, the solutions to the equation \(x^2 = 121\) are \(x = \pm 11\).

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. In our given problem, the term \(\sqrt{121}\) is an example of a radical expression, which denotes the square root of 121. A radical expression can occasionally result in a non-integer answer, which cannot be simplified to a whole number.

In such cases, it's necessary to leave the answer in radical form, like \(\sqrt{2}\), instead of converting to a decimal, as radical form is more precise. Radical expressions allow us to work with and simplify roots without approximation, maintaining the exactness required in mathematics.

No Real Solution

When working with equations that involve square roots, we may encounter situations where no real solution exists. This occurs when we attempt to take the square root of a negative number. In standard real number arithmetic, the square root of a negative number is not defined, as no real number multiplied by itself will yield a negative result.

For example, in an equation like \(x^2 = -1\), if we follow the steps to solve for \(x\), we would look for a number that squares to -1, which does not exist within the set of real numbers. Hence, we would declare there is no real solution.

Isolating Variables

Isolating the variable is a crucial step in solving equations. It means rewriting the equation in such a way that the variable we want to solve for is on one side of the equation by itself. In the equation \(x^2 = 121\), the variable \(x\) is squared and we need to isolate \(x\) to find its value.

To do this, we use inverse operations. Since \(x\) is squared, we take the square root of both sides, which is the inverse of squaring, to isolate \(x\). When the variable is isolated, we can see the solution(s) more clearly. In our case, after isolating \(x\), we found that \(x = \pm 11\).