Solve The Equation:${ 9x^2 - 25 = 0 }$

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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 $9x^2 - 25 = 0$. This equation can be solved using various methods, including factoring, completing the square, and the quadratic formula. We will explore each of these methods in detail and provide step-by-step solutions.

Factoring the Quadratic Equation

One of the simplest methods to solve a quadratic equation is by factoring. Factoring involves expressing the quadratic equation as a product of two binomials. To factor the equation $9x^2 - 25 = 0$, we need to find two numbers whose product is $-225$ and whose sum is $0$. These numbers are $15$ and $-15$, since $15 \times -15 = -225$ and $15 + (-15) = 0$.

We can now rewrite the equation as $(3x - 5)(3x + 5) = 0$. This is a factored form of the original equation, and it allows us to find the solutions by setting each factor equal to zero.

Solving for x using Factoring

To solve for $x$, we set each factor equal to zero and solve for $x$. We have two equations:

$3x - 5 = 0$

$3x + 5 = 0$

Solving the first equation, we get:

$3x = 5$

$x = \frac{5}{3}$

Solving the second equation, we get:

$3x = -5$

$x = -\frac{5}{3}$

Therefore, the solutions to the equation $9x^2 - 25 = 0$ are $x = \frac{5}{3}$ and $x = -\frac{5}{3}$.

Completing the Square

Another method to solve a quadratic equation is by completing the square. This method involves rewriting the quadratic equation in a perfect square form, which allows us to find the solutions easily.

To complete the square, we need to add and subtract a constant term to the equation. In this case, we add and subtract $(\frac{b}{2})^2$ to the equation, where $b$ is the coefficient of the linear term.

For the equation $9x^2 - 25 = 0$, we have $b = 0$, since there is no linear term. Therefore, we add and subtract $0^2 = 0$ to the equation.

We can now rewrite the equation as:

$9x^2 - 25 + 0 = 0$

$9x^2 = 25$

$x^2 = \frac{25}{9}$

Taking the square root of both sides, we get:

$x = \pm \sqrt{\frac{25}{9}}$

$x = \pm \frac{5}{3}$

Therefore, the solutions to the equation $9x^2 - 25 = 0$ are $x = \frac{5}{3}$ and $x = -\frac{5}{3}$.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is a general formula that can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$.

The quadratic formula is given by:

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

For the equation $9x^2 - 25 = 0$, we have $a = 9$, $b = 0$, and $c = -25$.

Substituting these values into the quadratic formula, we get:

$x = \frac{-0 \pm \sqrt{0^2 - 4(9)(-25)}}{2(9)}$

$x = \frac{0 \pm \sqrt{0 + 900}}{18}$

$x = \frac{0 \pm \sqrt{900}}{18}$

$x = \frac{0 \pm 30}{18}$

Therefore, the solutions to the equation $9x^2 - 25 = 0$ are $x = \frac{30}{18}$ and $x = -\frac{30}{18}$.

Simplifying these fractions, we get:

$x = \frac{5}{3}$

$x = -\frac{5}{3}$

Therefore, the solutions to the equation $9x^2 - 25 = 0$ are $x = \frac{5}{3}$ and $x = -\frac{5}{3}$.

Conclusion

In this article, we have explored three methods to solve the quadratic equation $9x^2 - 25 = 0$. We have used factoring, completing the square, and the quadratic formula to find the solutions.

The solutions to the equation are $x = \frac{5}{3}$ and $x = -\frac{5}{3}$. These solutions can be verified by substituting them back into the original equation.

We hope that this article has provided a clear and concise explanation of how to solve quadratic equations. We have used a step-by-step approach to make the solutions easy to follow.

Quadratic equations are an important concept in mathematics, and they have many real-world applications. We hope that this article has inspired you to learn more about quadratic equations and their applications.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we have explored three methods to solve the quadratic equation $9x^2 - 25 = 0$.

We have used factoring, completing the square, and the quadratic formula to find the solutions. The solutions to the equation are $x = \frac{5}{3}$ and $x = -\frac{5}{3}$.

We hope that this article has provided a clear and concise explanation of how to solve quadratic equations. We have used a step-by-step approach to make the solutions easy to follow.

Quadratic equations are an important concept in mathematics, and they have many real-world applications. We hope that this article has inspired you to learn more about quadratic equations and their applications.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Solving Quadratic Equations" by Khan Academy
• [3] "Quadratic Formula" by Wolfram MathWorld

Glossary

• Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Factoring: Expressing a quadratic equation as a product of two binomials.
• Completing the Square: Rewriting a quadratic equation in a perfect square form.
• Quadratic Formula: A general formula for solving quadratic equations of the form $ax^2 + bx + c = 0$.

About the Author

The author of this article is a mathematics enthusiast who has a passion for teaching and learning. They have a strong background in mathematics and have taught various mathematics courses to students of all ages.

The author is committed to providing high-quality content that is easy to understand and fun to read. They believe that mathematics should be accessible to everyone, and they strive to make complex concepts simple and intuitive.

The author has written several articles on mathematics and has a strong online presence. They are active on social media and engage with their readers to provide feedback and answer questions.

The author's goal is to inspire a love of mathematics in their readers and to provide them with the tools and resources they need to succeed in their mathematical journey.

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 this article, we will answer some of the most frequently asked questions about 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: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. We will explore each of these methods in detail in this article.

Q: What is the quadratic formula?

A: The quadratic formula is a general formula for solving quadratic equations of 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 substitute the values of $a$, $b$, and $c$ into the formula. Then, you need to simplify the expression and solve for $x$.

Q: What is the difference between factoring and completing the square?

A: Factoring involves expressing a quadratic equation as a product of two binomials, while completing the square involves rewriting a quadratic equation in a perfect square form.

Q: When should I use factoring and when should I use completing the square?

A: You should use factoring when the quadratic equation can be easily factored, and you should use completing the square when the quadratic equation cannot 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:

• Not checking the solutions to see if they are valid
• Not simplifying the expression before solving for $x$
• Not using the correct method to solve the equation

Q: How do I check if my solutions are valid?

A: To check if your solutions are valid, you need to substitute the solutions back into the original equation and see if they are true.

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

A: Quadratic equations have many 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 bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems.

Q: How can I practice solving quadratic equations?

A: You can practice solving quadratic equations by working through examples and exercises in a textbook or online resource. You can also try solving quadratic equations on your own and then check your solutions to see if they are correct.

Q: What are some resources for learning more about quadratic equations?

A: Some resources for learning more about quadratic equations include:

• Textbooks: There are many textbooks available that cover quadratic equations in detail.
• Online resources: There are many online resources available that provide tutorials and examples on quadratic equations.
• Video lectures: There are many video lectures available that cover quadratic equations in detail.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have many real-world applications. In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope that this article has provided you with a better understanding of quadratic equations and how to solve them.

Final Thoughts

Quadratic equations are an important concept in mathematics, and they have many real-world applications. We hope that this article has inspired you to learn more about quadratic equations and how to solve them.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Solving Quadratic Equations" by Khan Academy
• [3] "Quadratic Formula" by Wolfram MathWorld

Glossary

• Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Factoring: Expressing a quadratic equation as a product of two binomials.
• Completing the Square: Rewriting a quadratic equation in a perfect square form.
• Quadratic Formula: A general formula for solving quadratic equations of the form $ax^2 + bx + c = 0$.

About the Author

The author of this article is a mathematics enthusiast who has a passion for teaching and learning. They have a strong background in mathematics and have taught various mathematics courses to students of all ages.

The author is committed to providing high-quality content that is easy to understand and fun to read. They believe that mathematics should be accessible to everyone, and they strive to make complex concepts simple and intuitive.

The author has written several articles on mathematics and has a strong online presence. They are active on social media and engage with their readers to provide feedback and answer questions.

The author's goal is to inspire a love of mathematics in their readers and to provide them with the tools and resources they need to succeed in their mathematical journey.