Question

\(z^{2}=-z^{2}\)

Short answer

z = 0

Step-by-step solution

Set Up the Equation

Start with the given equation: \[z^{2} = -z^{2}\]

Combine Like Terms

Move all terms to one side of the equation to set it to zero. Add \(z^{2}\) to both sides: \[z^{2} + z^{2} = 0\]

Simplify

Combine the terms on the left side: \[2z^{2} = 0\]

Solve for z

Divide both sides of the equation by 2: \[z^{2} = 0\] Now take the square root of both sides: \[z = 0\]

Key concepts

Equation Solving

Solving equations is a foundational skill in algebra. The goal is to find the value of the variable that makes the equation true.
For example, in the equation given: \(z^{2} = -z^{2}\), you want to isolate the variable z to find its value.
Remember these steps:

• Move all terms to one side: This makes it easier to simplify the equation.
• Simplify the equation: Combine like terms to get a simpler expression to work with.
• Isolate the variable: Perform operations to solve for the unknown.

These steps can help you solve more complex equations as well.

Combining Like Terms

Combining like terms is a method to simplify equations. This means adding or subtracting terms with the same variable and exponent.
In our exercise, we combined the terms \(z^{2} + z^{2}\) to get \(2z^{2} = 0\).
When you have terms like \(3x\) and \(5x\), combining them gives \(8x\). Similarly, \(2y^{2} - y^{2}\) simplifies to \(y^{2}\).
This step is crucial before isolating the variable because it makes the equation simpler and easier to solve. Remember: always work with like terms to make the algebra easier.

Square Roots

Taking the square root of both sides is a common method for solving equations that involve squares.
In our exercise, after simplifying to \(z^{2} = 0\), we took the square root of both sides to find \(z\).
Remember: The square root function reverses the squaring operation. The square root of \(a^{2}\) is \(a\). For example, \(\sqrt{16} = 4\) because \(4^{2} = 16\).

• When you take the square root of both sides, don’t forget about the \(\backslash \pm\) symbol for potential negative roots, but here it's not applicable since 0 has only one square root, which is 0.
• Square roots help to simplify and solve equations where the variable is squared. It’s a powerful tool in algebra that opens the door to solving quadratic equations.