Solve For $z$.$z^2 = 0.64$Enter Your Answer In The Box.$ Z = ± 0.8 Z = \pm 0.8 Z = ± 0.8 [/tex]

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 delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will focus on the equation $z^2 = 0.64$ and explore the different methods of solving it.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula can be used to solve any quadratic equation in the form $ax^2 + bx + c = 0$. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the equation.

Solving the Equation $z^2 = 0.64$

Now, let's apply the quadratic formula to solve the equation $z^2 = 0.64$. We can rewrite the equation as $z^2 - 0.64 = 0$, which is in the form $ax^2 + bx + c = 0$. In this case, $a = 1$, $b = 0$, and $c = -0.64$.

Substituting these values into the quadratic formula, we get:

$$z = \frac{-0 \pm \sqrt{0^2 - 4(1)(-0.64)}}{2(1)}$$

Simplifying the expression, we get:

$$z = \frac{0 \pm \sqrt{2.56}}{2}$$

$$z = \frac{0 \pm 1.6}{2}$$

Therefore, we have two possible solutions:

$$z = \frac{0 + 1.6}{2} = 0.8$$

$$z = \frac{0 - 1.6}{2} = -0.8$$

Conclusion

In this article, we have explored the concept of quadratic equations and provided a step-by-step guide on how to solve them. We have applied the quadratic formula to solve the equation $z^2 = 0.64$ and obtained two possible solutions: $z = 0.8$ and $z = -0.8$. Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike.

Tips and Tricks

• When solving quadratic equations, make sure to identify the values of $a$, $b$, and $c$ in the equation.
• Use the quadratic formula to solve quadratic equations in the form $ax^2 + bx + c = 0$.
• Simplify the expression under the square root to obtain the solutions.
• Check your solutions by plugging them back into the original equation.

Common Mistakes

• Failing to identify the values of $a$, $b$, and $c$ in the equation.
• Using the quadratic formula incorrectly.
• Failing to simplify the expression under the square root.
• Not checking the solutions by plugging them back into the original equation.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

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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 is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, the quadratic formula, and graphing. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations in the form $ax^2 + bx + c = 0$. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, substitute these values into the quadratic formula and simplify the expression under the square root.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to solve the equation.

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations in the form $ax^2 + bx + c = 0$. However, it may not be the most efficient method for solving some equations, especially those that can be easily factored.

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

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

• Failing to identify the values of $a$, $b$, and $c$ in the equation.
• Using the quadratic formula incorrectly.
• Failing to simplify the expression under the square root.
• Not checking the solutions by plugging them back into the original equation.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, quadratic equations can be used to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Q: What are some advanced topics in quadratic equations?

A: Some advanced topics in quadratic equations include:

• Quadratic equations with complex roots.
• Quadratic equations with irrational roots.
• Quadratic equations with multiple solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the step-by-step guide provided in this article, you can solve quadratic equations with ease. Remember to identify the values of $a$, $b$, and $c$ in the equation, use the quadratic formula, simplify the expression under the square root, and check your solutions by plugging them back into the original equation.