Identify The Solution(s) Of $z^2 - 4 = 0$.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the quadratic equation $z^2 - 4 = 0$, which is a classic example of a quadratic equation. We will break down the solution process into manageable steps, making it easy to understand and follow.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, z) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants. In our equation, $z^2 - 4 = 0$, we can rewrite it as $z^2 - 4 = 0$, where a = 1, b = 0, and c = -4.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our equation, $z^2 - 4 = 0$, we can plug in the values of a, b, and c into the quadratic formula:

z=βˆ’0Β±02βˆ’4(1)(βˆ’4)2(1)z = \frac{-0 \pm \sqrt{0^2 - 4(1)(-4)}}{2(1)}

Simplifying the Quadratic Formula

Now, let's simplify the quadratic formula:

z=βˆ’0Β±0+162z = \frac{-0 \pm \sqrt{0 + 16}}{2}

z=βˆ’0Β±162z = \frac{-0 \pm \sqrt{16}}{2}

z=βˆ’0Β±42z = \frac{-0 \pm 4}{2}

Identifying the Solutions

Now that we have simplified the quadratic formula, we can identify the solutions:

z=βˆ’0+42z = \frac{-0 + 4}{2}

z=42z = \frac{4}{2}

z=2z = 2

z=βˆ’0βˆ’42z = \frac{-0 - 4}{2}

z=βˆ’42z = \frac{-4}{2}

z=βˆ’2z = -2

Conclusion

In this article, we have solved the quadratic equation $z^2 - 4 = 0$ using the quadratic formula. We have broken down the solution process into manageable steps, making it easy to understand and follow. We have identified the solutions as z = 2 and z = -2. This is a classic example of a quadratic equation, and we hope that this article has provided a clear and concise explanation of the solution process.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. It requires a deep understanding of algebraic concepts and the ability to apply them to real-world problems. In this article, we have provided a step-by-step guide to solving the quadratic equation $z^2 - 4 = 0$. We hope that this article has been helpful in providing a clear and concise explanation of the solution process.

Additional Resources

For further reading on quadratic equations, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, z) is two.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What are the solutions to the quadratic equation $z^2 - 4 = 0$?

A: The solutions to the quadratic equation $z^2 - 4 = 0$ are z = 2 and z = -2.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we provided a step-by-step guide to solving the quadratic equation $z^2 - 4 = 0$. In this article, we will answer some of the most frequently asked questions about quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, z) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Alternatively, you can also factor the quadratic equation, complete the square, or use a quadratic equation solver.

Q: What are the solutions to the quadratic equation $z^2 - 4 = 0$?

A: The solutions to the quadratic equation $z^2 - 4 = 0$ are z = 2 and z = -2.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is given by $b^2 - 4ac$. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is a value that determines the number of solutions to a quadratic equation. It is given by $b^2 - 4ac$.

Q: How do I use the discriminant to determine the number of solutions to a quadratic equation?

A: To use the discriminant to determine the number of solutions to a quadratic equation, you can follow these steps:

  1. Calculate the discriminant using the formula $b^2 - 4ac$.
  2. If the discriminant is positive, the equation has two distinct solutions.
  3. If the discriminant is zero, the equation has one repeated solution.
  4. If the discriminant is negative, the equation has no real solutions.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$, where a and b are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the difference between a quadratic equation and a polynomial equation?

A: A quadratic equation is a polynomial equation of degree two, while a polynomial equation is a general term that refers to any equation that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, where $a_n \neq 0$ and $n$ is a positive integer.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, you can use various methods, including factoring, the quadratic formula, and numerical methods.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope that this article has provided a clear and concise explanation of the concepts and methods involved in solving quadratic equations.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. It requires a deep understanding of algebraic concepts and the ability to apply them to real-world problems. In this article, we have provided a comprehensive guide to solving quadratic equations, including the quadratic formula, factoring, and the discriminant.

Additional Resources

For further reading on quadratic equations, we recommend the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, z) is two.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Alternatively, you can also factor the quadratic equation, complete the square, or use a quadratic equation solver.

Q: What are the solutions to the quadratic equation $z^2 - 4 = 0$?

A: The solutions to the quadratic equation $z^2 - 4 = 0$ are z = 2 and z = -2.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is given by $b^2 - 4ac$. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.