Solve the radical equation. Check for extraneous solutions.
$a=\sqrt{7 a-6}$
$a=\sqrt{7 a-6}^2$ Step 1: Square to get rid of the square root.
$-7 a-7 a$ Step 2: Subtract $7 a$ from both sides.
$\begin{aligned}
\frac{-6 a}{-6} & =\frac{-6}{-6} \\
a & =1
\end{aligned}$
Step 3: Divide by -6 .
The above problem was solved INCORRECTLY. Describe the mistake that was made in this problem.
A. Forgot to square the a in step 1.
B. Should have added 7a instead of subtracting in step 2.
C. Should have added 6 instead of dividing in step 3
Answer (1)
The mistake in the provided solution was that 'a' was not squared when both sides of the equation were squared. The correct equation after squaring both sides should have been a 2 = 7 a − 6 . The correct solutions are a = 1 and a = 6 . The mistake was forgetting to square the a in step 1.
Explanation
Analyzing the Incorrect Solution Let's analyze the given solution to the radical equation a = 7 a − 6 . The provided solution incorrectly solves the equation. We need to identify the mistake.
Identifying the First Mistake The first step in the provided solution is to square both sides. However, only the square root was squared, and not the 'a' on the left side of the equation. The correct first step should be: a 2 = ( 7 a − 6 ) 2 a 2 = 7 a − 6
Rearranging the Equation The next step is to rearrange the equation into a standard quadratic form by subtracting 7 a and adding 6 to both sides: a 2 − 7 a + 6 = 0
Factoring the Quadratic Now, we factor the quadratic equation: ( a − 6 ) ( a − 1 ) = 0
Finding Potential Solutions Solving for a , we get two possible solutions: a = 6 or a = 1
Checking for Extraneous Solutions We need to check for extraneous solutions by substituting these values back into the original equation a = 7 a − 6 .
For a = 6 :
6 = 7 ( 6 ) − 6 = 42 − 6 = 36 = 6 . This solution is valid. For a = 1 :
1 = 7 ( 1 ) − 6 = 7 − 6 = 1 = 1 . This solution is also valid.
Identifying the Error Comparing our correct steps with the provided solution, the mistake was in Step 1 where 'a' was not squared. The equation should have been a 2 = 7 a − 6 , not a = 7 a − 6 .
Examples
Radical equations appear in various fields, such as physics and engineering, when dealing with relationships involving square roots or other radicals. For example, when calculating the velocity of an object in free fall or determining the dimensions of a container with a specific volume, radical equations may arise. Understanding how to correctly solve these equations is crucial for accurate modeling and problem-solving in these disciplines. By mastering the techniques for solving radical equations, students can apply these skills to real-world scenarios and gain a deeper appreciation for the power of mathematics in describing and predicting physical phenomena.
