Question

Hazeem states that the equations \(\sqrt{x^{2}}=9\) and \((\sqrt{x})^{2}=9\) have the same solution. Is he correct? Justify your answer.

Short answer

No, Hazeem is not correct. The first equation has solutions \(x=9\) and \(x=-9\), while the second equation has only one solution, \(x=9\).

Step-by-step solution

Analyze the first equation

The first equation is \ \(\sqrt{x^{2}}=9\). To solve this, we recognize that the square root and the square are inverse operations. Therefore, the equation simplifies to \(x=9\) or \(x=-9\). So, the solutions to the first equation are \(x=9\) and \(x=-9\).

Analyze the second equation

The second equation is \ \((\sqrt{x})^{2}=9\). To solve this, notice that squaring the square root cancels out and we get \(x=9\). Therefore, the solution to the second equation is only \(x=9\).

Compare the solutions

The solutions to the first equation are \(x=9\) and \(x=-9\), while the solution to the second equation is only \(x=9\). Thus, the two equations do not have the same solutions.

Conclusion

Hazeem is not correct because \(\sqrt{x^{2}}=9\) has solutions \(x=9\) and \(x=-9\), while \((\sqrt{x})^{2}=9\) has only one solution, \(x=9\).

Key concepts

Inverse Operations

Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations because if you add a number and then subtract the same number, you return to your original number. In the context of equations, two common inverse operations are squaring and taking the square root. When you square a number and then take the square root, you return to your original number. This principle helps simplify the expression in the given equations. In \(\textstyle\text{√}{(x^2)}=9\), we use the inverse operations of squaring and taking the square root to solve for x.
Recognizing inverse operations helps make complex calculations more straightforward.

Square Root

The square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(\textstyle{3^2 = 9}\). However, there is an important nuance: both 3 and -3 are square roots of 9 because \(\textstyle{(-3)^2 = 9}\) too. In the equation \(\textstyle\text{√}{(x^2)}=9\), applying the square root to \(\textstyle x^2\) gives both positive and negative values.
On the other hand, in \(\textstyle(\text{√}{x})^2=9\), we first take the square root of x and then square it, which cancels out the square root, directly giving us x. This distinction is crucial for understanding why the solutions differ in the two equations.

Solving Equations

Solving equations involves finding all values of the variable that make the equation true. For the first equation, \(\textstyle\text{√}{(x^2)}=9\), we use the fact that square root and square are inverse operations, resulting in \(\textstyle x = 9\) or \(\textstyle x = -9\). This means both positive and negative roots are considered.
For the second equation, \(\textstyle(\text{√}{x})^2=9\), the operations simplify to \(\textstyle x = 9\) because squaring the square root of a number gives us back the original number, implying there's only one solution.
Understanding these operational differences is key to correctly solving the equations.

Comparison of Solutions

For this exercise, comparing solutions helps us understand that not all seemingly similar equations have the same solutions. From the first equation, \(\textstyle\text{√}{(x^2)}=9\), we found the solutions \(\textstyle x = 9\) and \(\textstyle x = -9\), due to the nature of the square root function.
Meanwhile, from the second equation, \(\textstyle(\text{√}{x})^2=9\), we found only \(\textstyle x = 9\), because squaring a square root results in a single non-negative value.
Thus, Hazeem's statement that the two equations have the same solutions is incorrect. Comparing solutions helps verify that seemingly similar equations can behave differently.