Question

select the correct answer from each drop-down menu.
two equations are shown.

equation a: ( x^2 = 49 )
equation b: ( x = sqrt{49} )

complete the sentence to compare the solutions to the equations when solving for ( x ).

the two equations have a different solution when solving for ( x ) because
squaring equation b on both sides yields the same solution as equation a.

Explanation:

Step1: Solve Equation A

To solve \( x^2 = 49 \), we take the square root of both sides. Remember that when we take the square root of a number, we get both a positive and a negative solution. So, \( x = \pm\sqrt{49} = \pm 7 \). Thus, the solutions for Equation A are \( x = 7 \) and \( x = -7 \).

Step2: Solve Equation B

For Equation B, \( x = \sqrt{49} \). The square root symbol (principal square root) denotes the non - negative square root. So, \( \sqrt{49}=7 \), which means the solution for Equation B is \( x = 7 \).

Step3: Compare the solutions

Equation A has two solutions (\( 7 \) and \( - 7\)), while Equation B has only one solution (\( 7 \)). So, the two equations have a different number of solutions. The reason is that the square root in Equation B represents the principal (non - negative) square root, giving only \( x = 7 \), whereas Equation A, when solved by taking the square root of both sides, gives both positive and negative roots. The statement "squaring equation B on both sides yields the same solution as equation A" is incorrect. If we square Equation B (\( x=\sqrt{49}\) or \( x = 7\)), we get \( x^{2}=49 \), which is Equation A. But the solution set of Equation A is \( \{7, - 7\} \) and the solution set of Equation B is \( \{7\} \). Squaring Equation B gives us the equation of A, but the solution of B is a subset of the solutions of A. The key difference is in the nature of the square root operation: the principal square root (in Equation B) gives one solution, while solving \( x^{2}=49 \) (Equation A) gives two solutions because a positive number has two square roots (positive and negative).

Answer:

The two equations have \(\boldsymbol{\text{a different}}\) number of solutions when solving for \(x\) because the solution to Equation A is \(x = 7\) or \(x=-7\) (two solutions) and the solution to Equation B is \(x = 7\) (one solution) (the given reason "squaring equation B on both sides yields the same solution as equation A" is incorrect as squaring gives the equation of A but not the same solution set). The correct reason is related to the principal square root in Equation B giving one solution and the square root operation in Equation A (when solving \(x^{2}=49\)) giving two solutions.