Glossary Checklist FAQ

Radicals

Before you can solve quadratic equations by completing the square, you must review the basics of using square roots. The \(\sqrt {}\) symbol is called a radical sign, and it is used to indicate the square root of the expression under the radical sign. The value or expression inside the radical is called the radicand. In Math 10C, the concern was only the positive (principal) square root, but in this course, you need to be aware of both the positive and negative square roots.

For example,

\(\begin{align}
 x^2 &= 9 \\
 \sqrt{x^2} &= \pm \sqrt 9  \\
 x &= \pm 3 \\
 \end{align}\)

The \(\sqrt {}\) symbol has the same meaning as \(\sqrt[2]{{}}\), but most textbooks and math literature do not include the \(2\), which indicates the index of the radical. If no index is shown, it is assumed to be \(2\), signifying a square root.

Solving Radical Equations

Negative numbers, when square rooted, produce Complex numbers (complex numbers are not studied at length in this course), not Real Numbers. If you find that you are in a position to take the square root of a negative number, you can conclude that there are no Real solution(s) to the equation.

Solve the following equations.

1. \(x^2 = 16\)
   
   
   Solution

   
   

   \(\begin{align}
    x^2 &= 16 \\
    \sqrt {x^2 } &= \pm \sqrt {16}  \\
    x &= \pm 4  \end{align}\)

   
   Recall that the square root of a squared number is equal to the number, or \(\sqrt {r^2 } = r\).
2. \(\left( {x - 1} \right)^2 = 25\)
   
   
   Solution

   \(\begin{align}
    \left( {x - 1} \right)^2 &= 25 \\
    \sqrt {\left( {x - 1} \right)^2 } &=  \pm \sqrt {25}  \\
    x - 1 &= \pm 5 \\
    x &= 1 \pm 5 \\
    \end{align}\)
   
   The \(\pm\) indicates that there are two possible answers,
   
   \(\begin{array}{l}
    x = 1 + 5 = 6 \\
    {\rm{or}} \\
    x = 1 - 5 = -4 \\
    \end{array}\)

Simplest Form

It is also important to write radicals in their simplest form. If the factors of the radicand are perfect squares, they can be removed from the radical by taking the square root, and then placing them in front of the radical sign.

To simplify a radical means to write it as a mixed radical, where the radicand does not have any perfect square factors.

Simplify the following radicals.

1. \(\sqrt {28} \)
   
   
   Solution

   \(\begin{align}
    \sqrt {28} &= \sqrt {4 \cdot 7}  \\
     &= 2\sqrt 7  \\
    \end{align}\)
2. \(\sqrt {800}\)
   
   
   Solution

   \(\begin{align}
    \sqrt {800} &= \sqrt {100 \cdot 8}  \\
     &= 10\sqrt 8  \\
     &= 10\sqrt {4 \cdot 2}  \\
     &= 20\sqrt 2   \end{align}\)

Exact Value

Unless a question asks for a rounded answer, leave the answer as a simplified radical. This is known as the exact value.

For example, \(\sqrt{2}\) is an exact value, whereas \(1.41\) is a rounded value.

If you require more review of how to take the square root of numbers and variables, please go to the Resources page.

Quadratic Equations

To solve a quadratic equation that is in vertex form, isolate the squared binomial term, and then take the square root of both sides. Remember when taking the square root that both a positive and a negative value are possible. Finally, solve and state the two possible solutions.