Yes, radical equations have the possibility of extraneous solutions.
Detailed response to a query
Yes, radical equations have the possibility of extraneous solutions. This is due to the fact that radical equations involve taking an even root of a number, which can result in both positive and negative solutions. However, when solving the equation, it is possible to introduce extra solutions that do not actually satisfy the original equation.
According to Walter Rudin, an American mathematician, “When we solve an equation containing radicals, we must examine extraneous solutions, which occur because raising both sides of the equation to an even power destroys information.”
It is important to carefully check the solutions obtained when solving radical equations, especially when raising both sides of the equation to an even power. Some interesting facts about radical equations and extraneous solutions include:
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Extraneous solutions can occur in various types of equations, including quadratic equations, radical equations, and equations with rational exponents.
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When solving a quadratic equation using the quadratic formula, it is possible to introduce extraneous solutions if the discriminant is negative.
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Radical equations can involve different types of radicals, including square roots, cube roots, and nth roots.
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The process of solving a radical equation typically involves isolating the radical expression and then squaring both sides of the equation.
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After squaring both sides of the equation, it is important to check the solutions obtained to ensure that they satisfy the original equation. If any extraneous solutions are found, they should be discarded.
A table summarizing some examples of radical equations and their possible extraneous solutions is shown below:
| Equation | Possible Extraneous Solution(s) |
|---|---|
| sqrt(x) = 2 | x = 4 (no extraneous solutions) |
| sqrt(x+1) = 3 | x = 8 (no extraneous solutions) |
| sqrt(x-5) = -2 | x = 1 (extraneous solution) |
| sqrt(x^2 – 9) = x – 3 | x = 6 (extraneous solution) |
| sqrt(2x+3) – sqrt(x+1) = 1 | x = 4 (no extraneous solutions) |
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When you square a radical equation you sometimes get a solution to the squared equation that is not a solution to the original equation. Such an equation is called an extraneous solution. Remember to always check your solutions in the original equation to discard the extraneous solutions.
When you square a radical equation you sometimes get a solution to the squared equation that is not a solution to the original equation. Such an equation is called an extraneous solution.
I looked it up, and came to know that such roots are called extraneous roots, and they occur when we raise both sides of a radical equation to an even power. And hence we must always plug the obtained values into the original equation in order to check whether they satisfy the parent equation or not.
Of course this doesn’t explain all of the extraneous solutions we get when solving radical equations, like when a linear graph has no chance of ever intersecting the radical at all.
Okay here the answer is true. You do sometimes get false answers. For example, the square root of X last night equals X plus two is a radical equation. And to solve it, I would square both sides. That’s the key point. When you square both sides, you can bring in extra solutions that don’t work. So here I square this and square this our caps, X plus eight equals X squared Plus four x plus four. And that will be arranged two, X squared plus three, X minus four equals zero. We arrange it. That will fact arise into X last four And X -1. So the answer has appeared to be negative for one. I’d like to hear them. We have roots of X-plus eight. Okay, Is equal to X-plus two. Let’s go If I plug in one that’s fine. Against the root of nine equals one plus two, correct plug in negative for what happens? I will get the root of four. That’s okay equals though -4 Plus two. You get two equals -2. Not correct. So this one, it doesn’t work. It’s extraneous. The truth is the answer. You do get extra class…
Video response to “Do radical equations have extraneous solutions?”
The video discusses the process of checking for extraneous solutions when solving radical equations. After solving the equation, each solution is plugged back into the original equation and tested to identify any solutions that do not fit. A sample equation is provided, and the steps to obtain and test solutions are demonstrated. The video highlights the importance of checking for extraneous solutions in various equation types, including radical, rational, absolute value, and logarithmic equations. The process of verifying solutions ensures accuracy and eliminates errors in the solution process.
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In this regard, Why do radical equations have extraneous solutions? Extraneous Solutions occur because squaring both sides of a square root equation results in 2 solutions (the positive and negative number). Therefore, one of those numbers will be an extraneous solution, or an extra solution which does not fulfill the original equation.
Similarly, What is an extraneous solution of a radical equation? In math, an extraneous solution is a solution that emerges during the process of solving a problem but is not actually a valid solution. You can only find out whether or not a solution is extraneous by plugging the solution back into the original equation.
Besides, Can rational equations have extraneous solutions?
For rational equations, extraneous solutions are values that cause any denominator in the original problem to be 0. Of course, when we have 0 in the denominator we have an expression that is undefined.
Additionally, Does radical equations always have real solutions?
Response will be: There will be no real number solutions. There are two key ideas that you will be using to solve radical equations. The first is that if a=b , then a2=b2 a 2 = b 2 . (This property allows you to square both sides of an equation and remain certain that the two sides are still equal.)
Besides, Is a radical equation a rational equation? Response to this: is a rational equation. Both radical and rational equations can have extraneous solutions, algebraic solutions that emerge as we solve the equations that do not satisfy the original equations. In other words, extraneous solutions seem like they’re solutions, but they aren’t.
Similarly, When do extraneous solutions of radical equations exist?
The response is: Extraneous solutions from radical equations exist whenever it is assumed that the principal root can return two values in one function. Show more… How’d you get 625-4 x 4 x 36/8?
Furthermore, What are extraneous solutions?
As an answer to this: Extraneous solutions are values that we get when solving equations that aren’t really solutions to the equation. In this video, we explain how and why we get extraneous solutions, by understanding the logic behind the process of solving equations. Want to join the conversation?
Moreover, Why is a rational equation extraneous? The response is: For example, if one of the solutions to a rational equation is 2 2 and the original equation contains the denominator x-2 x −2, then the solution 2 2 is extraneous because 2-2=0 2−2 = 0, and we cannot divide by 0 0. Solve the rational equation as outlined above. Substitute the solution (s) into the original equation.
Is a radical equation a rational equation?
In reply to that: is a rational equation. Both radical and rational equations can have extraneous solutions, algebraic solutions that emerge as we solve the equations that do not satisfy the original equations. In other words, extraneous solutions seem like they’re solutions, but they aren’t.
Simply so, Do we get extraneous solutions when solving equations?
Answer to this: Closed 4 months ago. We all know that wemight get extraneous solutions when solving certain equationslike rational equations and equations with radicals. In high school, teachers tell students that they should always check the solutions they obtained when solving equations like these because extraneous roots might appear.
How do you solve multiple terms under a radical? In the example above, only the variable x was underneath the radical. Sometimes you will need to solve an equation that contains multiple terms underneath a radical. Follow the same steps to solve these, but pay attention to a critical point: square both sides of an equation, not individual terms.
Considering this, Is x equals negative 3 an extraneous solution? Answer will be: X equals negative three is not a solution to this, but it is a solution for this and it is a solution to this quadratic right over here. So D equals two makes X equal negative three, an extraneous solution for this equation.