Which of the following is a radical equation?
Radical equations are mathematical expressions that involve at least one radical, which is a symbol representing the root of a number. These equations are often used to solve problems related to geometry, physics, and other fields. Identifying a radical equation is crucial in order to apply the appropriate mathematical techniques to find its solution. In this article, we will explore some examples of radical equations and discuss their characteristics.
Radical equations can be categorized into different types based on the presence of square roots, cube roots, or higher-order roots. Let’s consider a few examples to understand this concept better.
Example 1: Square Root Equation
Given the equation:
√(x + 3) = 4
This equation involves a square root, making it a radical equation. To solve it, we can square both sides to eliminate the radical:
(√(x + 3))^2 = 4^2
x + 3 = 16
Subtracting 3 from both sides, we get:
x = 13
Thus, the solution to the equation is x = 13.
Example 2: Cube Root Equation
Consider the equation:
∛(x – 2) = 3
This equation contains a cube root, which makes it a radical equation. To solve it, we can cube both sides to eliminate the radical:
(∛(x – 2))^3 = 3^3
x – 2 = 27
Adding 2 to both sides, we obtain:
x = 29
Hence, the solution to the equation is x = 29.
Example 3: Higher-Order Root Equation
Suppose we have the equation:
√(2x – 5) = √(x + 1)
This equation involves both square roots, indicating that it is a radical equation. To solve it, we can square both sides to eliminate the radicals:
(√(2x – 5))^2 = (√(x + 1))^2
2x – 5 = x + 1
Subtracting x from both sides, we get:
x = 6
So, the solution to the equation is x = 6.
In conclusion, identifying which of the following is a radical equation is essential for solving mathematical problems involving roots. By understanding the different types of radical equations and applying the appropriate techniques, we can find the solutions to these equations effectively.
