Equations with Radicals

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A series of free, online Basic Algebra Lessons.

In this lesson, we will learn

  • how to solve radical equations
  • how to graph radical equations using a table
  • how to graph radical equations using shifts
  • how to estimate square roots
  • how to find or simplify cube roots

Solving Radical Equations

Radical equations are equations that have square root terms. When solving radical equations, try to isolate the radical expression on one side of the equation and then square both sides (it is the inverse operation of taking a square root). If there are two or more terms on the opposite side of the equation, remember to draw parenthesis around that expression before squaring
Solving Radical Equations 1

Solving Radical Equations 2

Solving Radical Equations 3

Solving Radical Equations 4

Graphing Radical Equations using a Table

One way to graph radical functions is to create a table of values and then plot the points. Before starting the table, first determine the domain of the function. Remember, the radical must be greater than or equal to zero. Once this lower limit for input (domain) values is established, create the table of values. When graphing radicals we plot the points in the coordinate plane.
Graphing Radical Equations using a Table

This video is a demonstration of graphing radical equations by first (1) finding the domain, and (2) using an x-y table.

Graphing Radical Equations using Shifts

When graphing radical equations using shifts, adding or subtracting a constant that is not in the radical will shift the graph up (adding) or down (subtracting). Adding or subtracting a constant that is in the radical will shift the graph left (adding) or right (subtracting). Multiplying a negative constant by the equation will reflect the graph over the x-axis. Multiplying by a number larger than one increases the y-values.
How to graph radical functions without making a table of values.

Graphing Radicals using Vertical/Horizontal shifts and scaling.

Estimating Square Roots

When a radical is not a perfect square (1, 4, 9, 16, …), estimating square roots is a valuable tool. When asked to estimate the value of a radical between two consecutive integers, find the two perfect squares that are slightly less and slightly more than the radicand. Also, remember that negative numbers do not have a real number square root.
How to find square roots of integers.

Estimating Square Roots to the Nearest Integer

Simplify Cube Roots

A square root is an exponent of one-half. A cube root is an exponent of one-third. Square roots of negative numbers do not have real number roots since the product of any real number and itself is positive. Cube roots do exist for negative numbers since the product of three negatives is a negative. Cube roots re-appear often in Geometry and in Algebra II.
Finding Cube Roots

Simplify cube root

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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