What Is The Solution Set Of $x = \sqrt{3x + 40}$?

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Introduction

In mathematics, solving equations involving square roots can be a challenging task. The equation $x = \sqrt{3x + 40}$ is a classic example of such an equation. In this article, we will explore the solution set of this equation, which involves finding the values of x that satisfy the equation.

Understanding the Equation

The given equation is $x = \sqrt{3x + 40}$. To solve this equation, we need to isolate the variable x. The first step is to square both sides of the equation to eliminate the square root. This gives us:

x2=3x+40x^2 = 3x + 40

Squaring Both Sides

Squaring both sides of the equation is a common technique used to eliminate square roots. However, it's essential to note that this step can introduce extraneous solutions, which are values of x that satisfy the squared equation but not the original equation.

Expanding and Rearranging

Expanding the squared equation gives us:

x2−3x−40=0x^2 - 3x - 40 = 0

This is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -3, and c = -40.

Factoring the Quadratic Equation

To solve the quadratic equation, we can try factoring it. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor the quadratic equation as:

(x−8)(x+5)=0(x - 8)(x + 5) = 0

Finding the Solutions

To find the solutions, we set each factor equal to zero and solve for x. This gives us:

x−8=0⇒x=8x - 8 = 0 \Rightarrow x = 8

x+5=0⇒x=−5x + 5 = 0 \Rightarrow x = -5

Checking the Solutions

Now that we have found the solutions, we need to check if they satisfy the original equation. Substituting x = 8 into the original equation gives us:

8=3(8)+408 = \sqrt{3(8) + 40}

8=24+408 = \sqrt{24 + 40}

8=648 = \sqrt{64}

8=88 = 8

This confirms that x = 8 is a solution to the original equation.

Substituting x = -5 into the original equation gives us:

−5=3(−5)+40-5 = \sqrt{3(-5) + 40}

−5=−15+40-5 = \sqrt{-15 + 40}

−5=25-5 = \sqrt{25}

−5=5-5 = 5

This shows that x = -5 is not a solution to the original equation.

Conclusion

In conclusion, the solution set of the equation $x = \sqrt{3x + 40}$ is x = 8. This is the only value of x that satisfies the equation.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with the right techniques and strategies, it's possible to find the solution set. In this article, we explored the solution set of the equation $x = \sqrt{3x + 40}$ and found that the only solution is x = 8.

Additional Resources

For more information on solving equations involving square roots, check out the following resources:

  • Mathway: A online math problem solver that can help you solve equations involving square roots.
  • Khan Academy: A free online learning platform that offers video lessons and practice exercises on solving equations involving square roots.
  • Wolfram Alpha: A computational knowledge engine that can help you solve equations involving square roots.

Frequently Asked Questions

  • Q: What is the solution set of the equation $x = \sqrt{3x + 40}$? A: The solution set is x = 8.
  • Q: How do I solve equations involving square roots? A: You can use techniques such as squaring both sides, factoring, and checking the solutions.
  • Q: What are extraneous solutions? A: Extraneous solutions are values of x that satisfy the squared equation but not the original equation.

Glossary

  • Square root: A mathematical operation that finds the number that, when multiplied by itself, gives a specified value.
  • Extraneous solution: A value of x that satisfies the squared equation but not the original equation.
  • Quadratic equation: A polynomial equation of degree two, in the form of ax^2 + bx + c = 0.

Introduction

Solving equations involving square roots can be a challenging task, but with the right techniques and strategies, it's possible to find the solution set. In this article, we will answer some frequently asked questions about solving equations involving square roots.

Q: What is the solution set of the equation $x = \sqrt{3x + 40}$?

A: The solution set is x = 8. This is the only value of x that satisfies the equation.

Q: How do I solve equations involving square roots?

A: You can use techniques such as squaring both sides, factoring, and checking the solutions. Squaring both sides involves multiplying both sides of the equation by itself, which eliminates the square root. Factoring involves expressing the quadratic equation as a product of two binomials. Checking the solutions involves substituting the solutions back into the original equation to verify that they are true.

Q: What are extraneous solutions?

A: Extraneous solutions are values of x that satisfy the squared equation but not the original equation. These solutions can arise when squaring both sides of the equation, and they must be checked to ensure that they are not part of the solution set.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, substitute the solutions back into the original equation. If the solution does not satisfy the original equation, it is an extraneous solution and must be discarded.

Q: What is the difference between a quadratic equation and an equation involving a square root?

A: A quadratic equation is a polynomial equation of degree two, in the form of ax^2 + bx + c = 0. An equation involving a square root is an equation that contains a square root, such as $x = \sqrt{3x + 40}$. While both types of equations can be solved using similar techniques, the presence of a square root can make the equation more challenging to solve.

Q: Can I use a calculator to solve equations involving square roots?

A: Yes, you can use a calculator to solve equations involving square roots. However, it's essential to check the solutions to ensure that they are not extraneous.

Q: What are some common mistakes to avoid when solving equations involving square roots?

A: Some common mistakes to avoid when solving equations involving square roots include:

  • Squaring both sides of the equation without checking for extraneous solutions
  • Failing to check the solutions against the original equation
  • Not considering the possibility of extraneous solutions
  • Not using the correct techniques, such as factoring or checking the solutions

Q: How do I know if an equation is a quadratic equation or an equation involving a square root?

A: To determine if an equation is a quadratic equation or an equation involving a square root, look for the presence of a square root. If the equation contains a square root, it is an equation involving a square root. If the equation does not contain a square root, it is a quadratic equation.

Q: Can I use the quadratic formula to solve equations involving square roots?

A: No, you cannot use the quadratic formula to solve equations involving square roots. The quadratic formula is used to solve quadratic equations, not equations involving square roots.

Q: What are some real-world applications of solving equations involving square roots?

A: Solving equations involving square roots has many real-world applications, including:

  • Physics: Solving equations involving square roots is used to calculate distances, velocities, and accelerations.
  • Engineering: Solving equations involving square roots is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Solving equations involving square roots is used in algorithms and data structures.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right techniques and strategies, it's possible to find the solution set. By understanding the concepts and techniques involved in solving equations involving square roots, you can apply them to real-world problems and become a more confident and proficient problem solver.

Final Thoughts

Solving equations involving square roots is an essential skill in mathematics and has many real-world applications. By mastering this skill, you can tackle complex problems and become a more confident and proficient problem solver.

Additional Resources

For more information on solving equations involving square roots, check out the following resources:

  • Mathway: A online math problem solver that can help you solve equations involving square roots.
  • Khan Academy: A free online learning platform that offers video lessons and practice exercises on solving equations involving square roots.
  • Wolfram Alpha: A computational knowledge engine that can help you solve equations involving square roots.

Frequently Asked Questions

  • Q: What is the solution set of the equation $x = \sqrt{3x + 40}$? A: The solution set is x = 8.
  • Q: How do I solve equations involving square roots? A: You can use techniques such as squaring both sides, factoring, and checking the solutions.
  • Q: What are extraneous solutions? A: Extraneous solutions are values of x that satisfy the squared equation but not the original equation.

Glossary

  • Square root: A mathematical operation that finds the number that, when multiplied by itself, gives a specified value.
  • Extraneous solution: A value of x that satisfies the squared equation but not the original equation.
  • Quadratic equation: A polynomial equation of degree two, in the form of ax^2 + bx + c = 0.