Solve $x - \sqrt{-6x + 27} = 0$.
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Introduction
Solving equations involving square roots can be challenging, especially when the equation contains a variable inside the square root. In this case, we have an equation with a square root term that involves a variable, and our goal is to solve for the variable x. The given equation is $x - \sqrt{-6x + 27} = 0$.
Step 1: Isolate the Square Root Term
To solve this equation, we need to isolate the square root term. We can do this by moving the x term to the other side of the equation. This gives us $\sqrt{-6x + 27} = x$.
Step 2: Square Both Sides of the Equation
Squaring both sides of the equation will help us eliminate the square root. This gives us $\left(\sqrt{-6x + 27}\right)^2 = x^2$, which simplifies to $-6x + 27 = x^2$.
Step 3: Rearrange the Equation
Now, we need to rearrange the equation to get all the terms on one side. This gives us $x^2 + 6x - 27 = 0$.
Step 4: Solve the Quadratic Equation
The equation we have now is a quadratic equation in the form of $ax^2 + bx + c = 0$. We can solve this equation using the quadratic formula or factoring. In this case, we can factor the equation as $(x + 9)(x - 3) = 0$.
Step 5: Find the Solutions
Now that we have factored the equation, we can find the solutions by setting each factor equal to zero. This gives us $x + 9 = 0$ and $x - 3 = 0$. Solving for x, we get $x = -9$ and $x = 3$.
Step 6: Check the Solutions
Before we can consider the solutions valid, we need to check if they satisfy the original equation. We can do this by plugging the solutions back into the original equation. For $x = -9$, we get $-9 - \sqrt{-6(-9) + 27} = -9 - \sqrt{27} = -9 - 3\sqrt{3} \neq 0$, which means that $x = -9$ is not a valid solution. For $x = 3$, we get $3 - \sqrt{-6(3) + 27} = 3 - \sqrt{9} = 3 - 3 = 0$, which means that $x = 3$ is a valid solution.
Conclusion
In this article, we solved the equation $x - \sqrt{-6x + 27} = 0$ using algebraic techniques. We isolated the square root term, squared both sides of the equation, rearranged the equation, and solved the quadratic equation. We found two solutions, but only one of them satisfied the original equation. The valid solution is $x = 3$.
Final Answer
The final answer is $\boxed{3}$.
Related Topics
- Solving equations involving square roots
- Quadratic equations
- Algebraic techniques for solving equations
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
Further Reading
- [1] "Solving Equations Involving Square Roots" by Math Open Reference
- [2] "Quadratic Equations" by Khan Academy
- [3] "Algebraic Techniques for Solving Equations" by Wolfram MathWorld
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Introduction
In our previous article, we solved the equation $x - \sqrt{-6x + 27} = 0$ using algebraic techniques. In this article, we will answer some frequently asked questions related to solving equations involving square roots.
Q: What is the first step in solving an equation involving a square root?
A: The first step in solving an equation involving a square root is to isolate the square root term. This means moving all the terms that are not inside the square root to the other side of the equation.
Q: How do I know if I can square both sides of the equation?
A: You can square both sides of the equation if the expression inside the square root is a perfect square or if the equation is in the form of $x = \sqrt{a}$, where a is a constant.
Q: What happens if I square both sides of the equation and get a negative number?
A: If you square both sides of the equation and get a negative number, it means that the original equation has no real solutions. This is because the square of any real number is always non-negative.
Q: Can I use the quadratic formula to solve an equation involving a square root?
A: Yes, you can use the quadratic formula to solve an equation involving a square root. However, you need to be careful when applying the quadratic formula, as it may not always give you the correct solution.
Q: How do I check if a solution satisfies the original equation?
A: To check if a solution satisfies the original equation, you need to plug the solution back into the original equation and see if it is true. If the solution satisfies the original equation, then it is a valid solution.
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 if the expression inside the square root is a perfect square.
- Not checking if the solution satisfies the original equation.
- Not considering complex solutions.
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, you need to be careful when using a calculator, as it may not always give you the correct solution.
Q: How do I know if an equation involving a square root has a real solution?
A: An equation involving a square root has a real solution if the expression inside the square root is non-negative. You can check this by plugging in a value for x and seeing if the expression inside the square root is non-negative.
Q: Can I use algebraic techniques to solve equations involving square roots with variables inside the square root?
A: Yes, you can use algebraic techniques to solve equations involving square roots with variables inside the square root. However, you need to be careful when applying these techniques, as they may not always give you the correct solution.
Conclusion
In this article, we answered some frequently asked questions related to solving equations involving square roots. We covered topics such as isolating the square root term, squaring both sides of the equation, checking solutions, and common mistakes to avoid.
Final Answer
The final answer is that solving equations involving square roots requires careful attention to detail and a thorough understanding of algebraic techniques.
Related Topics
- Solving equations involving square roots
- Quadratic equations
- Algebraic techniques for solving equations
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
Further Reading
- [1] "Solving Equations Involving Square Roots" by Math Open Reference
- [2] "Quadratic Equations" by Khan Academy
- [3] "Algebraic Techniques for Solving Equations" by Wolfram MathWorld