# Solving Radical Equations Worksheet, Definition, and Examples

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### Key Points about Solving Radical Equations

- Radical equations involve a radical of an expression containing a variable.
- To solve a radical equation, isolate the radical on one side of the equation, raise both sides to a power that will eliminate the radical, and solve the equation.
- Radical equations are commonly used in algebra and calculus, as well as in real-life situations such as medicine and science.

## What are Radical Equations?

Solving Radical Equations is an important concept in mathematics that involves equations containing radicals. A radical equation is an equation that involves a radical of an expression containing a variable. These equations can be solved by isolating the radical on one side of the equation, raising both sides to a power that will eliminate the radical, and solving the equation.

Radical Equations are commonly found in algebra and calculus. They are used to solve problems that involve variables and exponents. They are also used in real-life situations such as calculating the amount of medication needed for a patient based on their body weight or determining the amount of time it takes for a radioactive substance to decay.

In this article, we will discuss how to solve radical equations, provide examples of radical equations, and explore the real-life applications of radical equations. We will also answer some frequently asked questions about radical equations.

**Common Core Standard:**

**How to Solve Radical Equations**

Radical equations are equations that involve radicals, which are expressions containing square roots, cube roots, and other roots. Solving these equations can be intimidating, but it is actually quite simple once you know the steps. Here are the steps to solve radical equations:

- Isolate the radical on one side of the equation: The first step is to isolate the radical on one side of the equation. This means moving all other terms to the other side of the equation. For example, if the equation is √(x+3) = 5, subtract 3 from both sides to get √(x+3) – 3 = 2.
- Square both sides of the equation: The next step is to square both sides of the equation. This will eliminate the radical on the left side of the equation. However, it will also introduce a new solution, so it is important to check the solutions later. Continuing with the example above, squaring both sides of the equation gives ( √(x+3) – 3 )^2 = 2^2, which simplifies to x+3 = 16.
- Solve for the variable: The next step is to solve for the variable. In the example above, subtracting 3 from both sides gives x = 13.
- Check the solutions: The final step is to check the solutions. This is important because squaring both sides of the equation can introduce extraneous solutions, which are solutions that do not actually satisfy the original equation. To check the solution, simply substitute it back into the original equation and see if it is true. In the example above, substituting x = 13 into the original equation gives √(13+3) = 5, which is true.

In summary, to solve radical equations, isolate the radical, square both sides of the equation, solve for the variable, and check the solutions.

**Radical Equations Definition**

A radical equation is a type of equation that contains a variable within a radical expression. The radical expression can be a square root, cube root, or any other root. The variable is typically located inside the radical, but it may also be located outside the radical.

Radical equations are unique because they involve an operation that is the inverse of the operation used to create the radical expression. For example, if a radical expression is created by taking the square root of a number, then the inverse operation is squaring the number.

Radical equations can be solved by isolating the radical expression and then applying the inverse operation to both sides of the equation. However, it is important to note that when solving radical equations, extraneous solutions may arise. These are solutions that do not satisfy the original equation and must be discarded.

Examples of radical equations include:

- √(x+3) = 5
- ³√(2x-1) = 4
- ∛(x²+3) = 7

It is important to note that radical equations can become more complex when multiple radical expressions are involved. In these cases, it may be necessary to use algebraic techniques such as factoring or substitution to simplify the equation before solving.

**Radical Equations Examples**

Here are some examples of how to solve radical equations:

**Example 1**

Solve the equation: √(x+7) – 4 = 0

To solve this equation, first add 4 to both sides: √(x+7) = 4

Then, square both sides to get rid of the radical: x+7 = 16

Finally, subtract 7 from both sides: x = 9

Therefore, the solution to the equation is x = 9.

**Example 2**

Solve the equation: √(x-3) + 5 = 7

To solve this equation, first subtract 5 from both sides: √(x-3) = 2

Then, square both sides to get rid of the radical: x-3 = 4

Finally, add 3 to both sides: x = 7

Therefore, the solution to the equation is x = 7.

**Example 3**

Solve the equation: √(2x+1) = 3

To solve this equation, square both sides to get rid of the radical: 2x+1 = 9

Then, subtract 1 from both sides: 2x = 8

Finally, divide both sides by 2: x = 4

Therefore, the solution to the equation is x = 4.

These examples demonstrate the basic steps involved in solving radical equations. It is important to remember to check for extraneous solutions and to always verify the solution by plugging it back into the original equation.

**Radical Equations in Real Life**

Radical equations are not only useful in mathematics, but they also have practical applications in real life. Here are a few examples of how radical equations are used in everyday situations:

**Kinetic Energy**

The kinetic energy of an object is given by the equation KE = 1/2mv², where m is the mass of the object and v is its velocity. If the velocity of an object is given in terms of a radical function, then the equation for kinetic energy becomes a radical equation. For example, if the velocity of an object is given by v = √(2gh), where g is acceleration due to gravity and h is the height of the object, then the equation for kinetic energy becomes KE = 1/2m(2gh).

**Electrical Circuits**

In electrical circuits, the current passing through a resistor is given by Ohm’s Law, which states that I = V/R, where I is the current, V is the voltage, and R is the resistance of the resistor. If the resistance of the resistor is given in terms of a radical function, then the equation for current becomes a radical equation. For example, if the resistance of a resistor is given by R = √(L/πr), where L is the length of the resistor and r is its radius, then the equation for current becomes I = V/√(L/πr).

**Sound Waves**

The frequency of a sound wave is given by the equation f = (1/2π)√(k/μ), where k is the stiffness of the medium and μ is its density. If the stiffness of the medium is given in terms of a radical function, then the equation for frequency becomes a radical equation. For example, if the stiffness of the medium is given by k = √(E/ρ), where E is the Young’s modulus of the medium and ρ is its density, then the equation for frequency becomes f = (1/2π)√(√(E/ρ)/μ).

Radical equations are used in various fields of science and engineering to model real-life situations. Understanding how to solve these equations is essential for solving problems in these fields.

**Solving Radical Equations FAQ**

**What is the process for solving radical equations with fractions?**

When solving radical equations with fractions, the first step is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD). Once the equation is free of fractions, the next step is to isolate the radical term on one side of the equation and then square both sides of the equation to eliminate the radical. After that, the equation can be solved by using basic algebraic techniques.

**How do you solve radical equations with two radicals?**

To solve radical equations with two radicals, the key is to isolate one of the radicals on one side of the equation and then square both sides of the equation to eliminate the radical. After that, the equation can be simplified and solved using basic algebraic techniques.

**What are some common mistakes to avoid when solving radical equations?**

Some common mistakes to avoid when solving radical equations include forgetting to check for extraneous solutions, failing to simplify the equation before solving it, and incorrectly applying the properties of exponents.

**How can I check my solutions for radical equations?**

To check solutions for radical equations, simply substitute the values of the variables back into the original equation and see if it is a true statement. If it is, then the solution is valid. If not, then it is an extraneous solution and should be discarded.

**What are some real-world applications of solving radical equations?**

Solving radical equations is useful in a variety of real-world applications, such as calculating the time it takes for a radioactive substance to decay to a certain level or determining the distance a sound wave travels before it becomes inaudible.

**What strategies can I use to solve radical equations more efficiently?**

Some strategies for solving radical equations more efficiently include simplifying the equation before solving it, factoring out perfect squares, and using substitution to simplify the equation.

**What is a radical equation with example?**

A radical equation is an equation that contains a radical expression. An example of a radical equation is √(x + 3) = 5.

**How do you write an equation as a radical?**

To write an equation as a radical, simply isolate the expression that is under the radical sign and then write it as a radical. For example, the equation x^2 = 9 can be written as x = ±√9.

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