Solve The Equation:${ 5z^2 - 80 = 0 }$

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on solving the quadratic equation $5z^2 - 80 = 0$.

Understanding the Equation

The given equation is $5z^2 - 80 = 0$. To solve this equation, we need to isolate the variable $z$. The first step is to add $80$ to both sides of the equation, which gives us $5z^2 = 80$. This equation is a quadratic equation in the form of $az^2 = b$, where $a = 5$ and $b = 80$.

Solving for $z$

To solve for $z$, we need to isolate the variable by dividing both sides of the equation by $5$. This gives us $z^2 = \frac{80}{5}$. Simplifying the right-hand side, we get $z^2 = 16$. Now, we need to find the square root of both sides of the equation to isolate $z$.

Finding the Square Root

Taking the square root of both sides of the equation, we get $z = \pm \sqrt{16}$. Simplifying the right-hand side, we get $z = \pm 4$. Therefore, the solutions to the equation $5z^2 - 80 = 0$ are $z = 4$ and $z = -4$.

Graphical Representation

To visualize the solutions, we can graph the equation $5z^2 - 80 = 0$. The graph of a quadratic equation is a parabola that opens upwards or downwards. In this case, the parabola opens upwards because the coefficient of $z^2$ is positive. The vertex of the parabola is at the point $(0, -80)$, and the axis of symmetry is the vertical line $z = 0$.

Real-World Applications

Quadratic equations have numerous real-world applications in various fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Conclusion

In conclusion, solving quadratic equations is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, we can solve the quadratic equation $5z^2 - 80 = 0$ and find the roots of the equation. The solutions to the equation are $z = 4$ and $z = -4$, and the graph of the equation is a parabola that opens upwards.

Additional Resources

For further reading and practice, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "College Algebra" by James Stewart
• Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Mathematica, Maple, MATLAB

Final Thoughts

Solving quadratic equations is a fundamental skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, we can solve the quadratic equation $5z^2 - 80 = 0$ and find the roots of the equation. We hope this article has provided a comprehensive guide to solving quadratic equations and has inspired readers to explore the world of mathematics.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve the quadratic equation $5z^2 - 80 = 0$. In this article, we will address some frequently asked questions and provide answers to help readers better understand quadratic equations.

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 is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable by using algebraic manipulations. The steps to solve a quadratic equation are:

1. Add or subtract the same value to both sides of the equation to isolate the variable.
2. Multiply or divide both sides of the equation by the same value to simplify the equation.
3. Use the quadratic formula to find the roots of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the roots of a quadratic equation. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of the variable that satisfy the equation. In other words, the roots of a quadratic equation are the values of $x$ that make the equation true.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program. The graph of a quadratic equation is a parabola that opens upwards or downwards.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous real-world applications in various fields such as physics, engineering, and economics. Some examples of applications of quadratic equations include:

• Modeling the trajectory of a projectile under the influence of gravity
• Describing the motion of a pendulum
• Finding the maximum or minimum value of a function

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most graphing calculators and computer programs have built-in functions to solve quadratic equations.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not following the order of operations
• Not simplifying the expression
• Not using the correct formula

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics that has numerous real-world applications. By understanding the basics of quadratic equations and using the quadratic formula, you can solve quadratic equations and find the roots of the equation. We hope this article has provided a comprehensive guide to quadratic equations and has inspired readers to explore the world of mathematics.

Additional Resources

For further reading and practice, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "College Algebra" by James Stewart
• Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Mathematica, Maple, MATLAB

Final Thoughts

Quadratic equations are a fundamental concept in mathematics that has numerous real-world applications. By understanding the basics of quadratic equations and using the quadratic formula, you can solve quadratic equations and find the roots of the equation. We hope this article has provided a comprehensive guide to quadratic equations and has inspired readers to explore the world of mathematics.