Explain The Method Used To Solve An Equation Containing Radi
Explain The Method Used To Solve An Equation Containing Radical Expres
Explain the method used to solve an equation containing radical expressions. Why is it necessary to check the solutions for such an equation? Provide an example – solve it and check its solutions. Given a function such as f ( x ) = 3 x – 7, what does it mean to say it is one-to-one? Provide an additional example for your classmates to test. (It need not be one-to-one).
Paper For Above instruction
The process of solving equations involving radical expressions requires systematic approaches to isolate the radical term, eliminate it, and then solve for the variable. These methods typically involve algebraic operations such as isolating the radical and then raising both sides of the equation to an power that removes the radical, followed by solving the resulting algebraic equation. Importantly, solutions obtained through this process must be verified in the original equation because the operations used—particularly raising both sides to an power—may introduce extraneous solutions that do not satisfy the initial equation.
The general method for solving an equation that contains a radical expression, such as a square root, involves the following steps:
1. Isolate the radical: Rearrange the equation to have the radical expression on one side and everything else on the other side.
2. Eliminate the radical: Raise both sides of the equation to a power that clears the radical. For example, if the radical is a square root, square both sides; if it is a cube root, cube both sides, and so on.
3. Solve the resulting algebraic equation: After removing the radical, solve the equation for the variable using standard algebraic methods.
4. Check solutions: Substitute the potential solutions back into the original radical equation to verify their validity. This step is critical because extraneous solutions can arise when both sides are raised to a power, especially if the radical expression was isolated improperly or if the domain restrictions are not considered.
For example, consider the radical equation:
\[
\sqrt{x + 3} = x - 1
\]
Step 1: Isolate the radical
The radical is already isolated on one side.
Step 2: Eliminate the radical
Square both sides:
\[
(\sqrt{x + 3})^2 = (x - 1)^2
\]
\[
x + 3 = (x - 1)^2
\]
\[
x + 3 = x^2 - 2x + 1
\]
Step 3: Solve the algebraic equation
Bring all terms to one side:
\[
0 = x^2 - 3x - 2
\]
Factor the quadratic:
\[
0 = (x - 2)(x - (-1))
\]
\[
0 = (x - 2)(x + 1)
\]
Solutions:
\[
x = 2,\quad x = -1
\]
Step 4: Check solutions in the original equation
For \(x=2\):
\[
\sqrt{2 + 3} = 2 - 1
\]
\[
\sqrt{5} \approx 2.236 \neq 1
\]
Since \(\sqrt{5} \neq 1\), \(x=2\) is an extraneous solution and is invalid.
For \(x=-1\):
\[
\sqrt{-1 + 3} = -1 - 1
\]
\[
\sqrt{2} \approx 1.414 \neq -2
\]
The right side is negative, but the left is positive; thus, \(x=-1\) does not satisfy the original equation.
Conclusion:
Neither solution is valid, so the original radical equation has no real solutions.
The necessity of checking solutions stems from the possibility of extraneous solutions introduced during the algebraic manipulation—particularly when raising both sides of the equation to a power. Not checking these solutions can lead to incorrect conclusions about the solutions of the equation.
Next, consider the function \(f(x) = 3x - 7\). Saying that this function is one-to-one (injective) means that for every pair of distinct inputs, the outputs are also distinct. In other words, if \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\). This property guarantees that the function passes the horizontal line test, and it ensures that each output corresponds to exactly one input.
For example, test whether \(f(x) = 3x - 7\) is one-to-one:
Suppose \(f(x_1) = f(x_2)\):
\[
3x_1 - 7 = 3x_2 - 7
\]
\[
3x_1 = 3x_2
\]
\[
x_1 = x_2
\]
Since the only way for \(f(x_1) = f(x_2)\) is if \(x_1 = x_2\), the function is one-to-one.
An additional example that is not one-to-one could be a quadratic function such as \(g(x) = x^2\). For instance, \(g(2) = 4\) and \(g(-2) = 4\). Here, two different inputs produce the same output, which violates the definition of a one-to-one function.
In conclusion, solving radical equations requires careful steps to eliminate radicals through exponentiation, followed by solution verification to avoid including extraneous solutions. The notion of a function being one-to-one is fundamental in understanding functions' invertibility and behavior, with linear functions like \(f(x) = 3x - 7\) inherently being one-to-one due to their constant rate of change.
References
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- Khan Academy. (2020). Solving radical equations. https://www.khanacademy.org/math/algebra/radical-equations
- Stewart, J. (2015). Calculus: Early Transcendental Mathematics. Cengage Learning.
- Singh, S. (2018). Basic Algebra. Pearson.
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