Solve For All Possible Values Of $x$.$\sqrt{8x+65}=x+10$

Feb 27, 2025 by ADMIN 57 views

**Solving Radical Equations: A Step-by-Step Guide**

Introduction

Radical equations are a type of algebraic equation that involves a variable under a radical sign. In this article, we will focus on solving radical equations of the form $\sqrt{ax+b}=cx+d$ , where $a$ , $b$ , $c$ , and $d$ are constants. We will use the given equation $\sqrt{8x+65}=x+10$ as an example to demonstrate the steps involved in solving radical equations.

Understanding Radical Equations

A radical equation is an equation that contains a variable under a radical sign. The radical sign is denoted by $\sqrt{}$ . In the given equation, the variable $x$ is under the radical sign. To solve this equation, we need to isolate the variable $x$ .

Step 1: Square Both Sides of the Equation

The first step in solving a radical equation is to square both sides of the equation. This will eliminate the radical sign and allow us to work with a polynomial equation.

$\sqrt{8x+65}=x+10$

Squaring both sides of the equation gives:

$8x+65=(x+10)^2$

Step 2: Expand the Right-Hand Side of the Equation

To expand the right-hand side of the equation, we need to use the formula $(a+b)^2=a^2+2ab+b^2$ .

$8x+65=x^2+20x+100$

Step 3: Simplify the Equation

Now that we have expanded the right-hand side of the equation, we can simplify the equation by combining like terms.

$x^2+20x+100-8x-65=0$

Simplifying the equation gives:

$x^2+12x+35=0$

Step 4: Factor the Quadratic Equation

The next step is to factor the quadratic equation. If the equation can be factored, we can find the solutions by setting each factor equal to zero.

$x^2+12x+35=(x+5)(x+7)$

Step 5: Set Each Factor Equal to Zero

Now that we have factored the quadratic equation, we can set each factor equal to zero and solve for $x$ .

$x+5=0$ or $x+7=0$

Solving for $x$ gives:

$x=-5$ or $x=-7$

Step 6: Check the Solutions

Before we can be sure that we have found the solutions to the equation, we need to check each solution by plugging it back into the original equation.

$\sqrt{8(-5)+65}=(-5)+10$

Simplifying the equation gives:

$\sqrt{15}=5$

Since $\sqrt{15}\neq 5$ , we can conclude that $x=-5$ is not a solution to the equation.

$\sqrt{8(-7)+65}=(-7)+10$

Simplifying the equation gives:

$\sqrt{1}=3$

Since $\sqrt{1}\neq 3$ , we can conclude that $x=-7$ is not a solution to the equation.

Conclusion

In this article, we have demonstrated the steps involved in solving radical equations. We used the given equation $\sqrt{8x+65}=x+10$ as an example to show how to square both sides of the equation, expand the right-hand side, simplify the equation, factor the quadratic equation, set each factor equal to zero, and check the solutions. We found that the solutions to the equation are $x=-5$ and $x=-7$ , but we also found that neither of these solutions is valid.

Common Mistakes to Avoid

When solving radical equations, there are several common mistakes to avoid. These include:

Squaring both sides of the equation without checking to see if the equation is a perfect square trinomial.
Expanding the right-hand side of the equation without checking to see if the equation is a perfect square trinomial.
Simplifying the equation without checking to see if the equation is a perfect square trinomial.
Factoring the quadratic equation without checking to see if the equation is a perfect square trinomial.
Setting each factor equal to zero without checking to see if the equation is a perfect square trinomial.
Checking the solutions without plugging them back into the original equation.

Real-World Applications

Radical equations have many real-world applications. Some examples include:

Physics: Radical equations are used to model the motion of objects under the influence of gravity.
Engineering: Radical equations are used to model the behavior of electrical circuits.
Computer Science: Radical equations are used to model the behavior of algorithms.

Final Thoughts

Introduction

Radical equations are a type of algebraic equation that involves a variable under a radical sign. In our previous article, we demonstrated the steps involved in solving radical equations. In this article, we will answer some common questions about solving radical equations.

Q: What is a radical equation?

A: A radical equation is an equation that contains a variable under a radical sign. The radical sign is denoted by $\sqrt{}$ . In the equation $\sqrt{8x+65}=x+10$ , the variable $x$ is under the radical sign.

Q: How do I solve a radical equation?

A: To solve a radical equation, you need to follow these steps:

Square both sides of the equation.
Expand the right-hand side of the equation.
Simplify the equation.
Factor the quadratic equation.
Set each factor equal to zero.
Check the solutions.

Q: Why do I need to square both sides of the equation?

A: Squaring both sides of the equation eliminates the radical sign and allows you to work with a polynomial equation. This makes it easier to solve the equation.

Q: What if the equation is not a perfect square trinomial?

A: If the equation is not a perfect square trinomial, you may need to use other methods to solve it. For example, you can use the quadratic formula or complete the square.

Q: How do I check the solutions?

A: To check the solutions, you need to plug each solution back into the original equation. If the solution satisfies the equation, then it is a valid solution.

Q: What if I get a negative value under the radical sign?

A: If you get a negative value under the radical sign, then the equation has no real solutions. This is because the square root of a negative number is not a real number.

Q: Can I use a calculator to solve radical equations?

A: Yes, you can use a calculator to solve radical equations. However, you need to be careful when using a calculator to solve radical equations. Make sure to check the solutions to ensure that they are valid.

Q: Are radical equations used in real-world applications?

A: Yes, radical equations are used in many real-world applications. Some examples include:

Physics: Radical equations are used to model the motion of objects under the influence of gravity.
Engineering: Radical equations are used to model the behavior of electrical circuits.
Computer Science: Radical equations are used to model the behavior of algorithms.

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

A: Some common mistakes to avoid when solving radical equations include:

Squaring both sides of the equation without checking to see if the equation is a perfect square trinomial.
Expanding the right-hand side of the equation without checking to see if the equation is a perfect square trinomial.
Simplifying the equation without checking to see if the equation is a perfect square trinomial.
Factoring the quadratic equation without checking to see if the equation is a perfect square trinomial.
Setting each factor equal to zero without checking to see if the equation is a perfect square trinomial.
Checking the solutions without plugging them back into the original equation.

Conclusion

In this article, we have answered some common questions about solving radical equations. We have also provided some tips and tricks for solving radical equations. By following these steps and avoiding common mistakes, you can solve radical equations with confidence.

Additional Resources

If you are interested in learning more about solving radical equations, here are some additional resources:

Khan Academy: Radical Equations
Mathway: Radical Equations
Wolfram Alpha: Radical Equations

Final Thoughts

Solving radical equations requires a step-by-step approach. By following these steps and avoiding common mistakes, you can solve radical equations with confidence. Remember to check the solutions to ensure that they are valid, and don't be afraid to use a calculator to help you solve the equation. With practice and patience, you can become proficient in solving radical equations.