Solve The Equation:${ 25x^2 - 16 = 0 }$

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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 $25x^2 - 16 = 0$.

Understanding the Equation

The given equation is $25x^2 - 16 = 0$. To solve this equation, we need to isolate the variable $x$. The first step is to add $16$ to both sides of the equation, which gives us $25x^2 = 16$. This equation is in the form of a quadratic equation, where the coefficient of $x^2$ is $25$, and the constant term is $16$.

Solving the Quadratic Equation

To solve the quadratic equation $25x^2 = 16$, we can use the following steps:

1. Divide both sides of the equation by 25: This gives us $x^2 = \frac{16}{25}$.
2. Take the square root of both sides of the equation: This gives us $x = \pm \sqrt{\frac{16}{25}}$.
3. Simplify the expression: We can simplify the expression by dividing the numerator and denominator by their greatest common divisor, which is $4$. This gives us $x = \pm \frac{4}{5}$.

The Solutions to the Equation

The solutions to the equation $25x^2 - 16 = 0$ are $x = \frac{4}{5}$ and $x = -\frac{4}{5}$. These are the values of $x$ that satisfy the equation.

Graphical Representation

We can represent the solutions to the equation graphically by plotting the graph of the equation. The graph of the equation $25x^2 - 16 = 0$ is a parabola that opens upwards, and it intersects the x-axis at the points $x = \frac{4}{5}$ and $x = -\frac{4}{5}$.

Real-World Applications

Quadratic equations have numerous real-world applications. For example, they are used in physics to describe the motion of objects under the influence of gravity, in engineering to design bridges and buildings, and in economics to model the behavior of markets.

Conclusion

In conclusion, solving quadratic equations is an essential skill in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, we can solve the quadratic equation $25x^2 - 16 = 0$ and find the solutions to the equation.

Additional Resources

For further reading on quadratic equations, 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 concept in mathematics, and it has numerous real-world applications. By mastering the skills outlined in this article, we can solve quadratic equations and apply them to real-world problems.

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 can use the following steps:

1. Divide both sides of the equation by the coefficient of $x^2$: This gives us $x^2 = \frac{c}{a}$.
2. Take the square root of both sides of the equation: This gives us $x = \pm \sqrt{\frac{c}{a}}$.
3. Simplify the expression: We can simplify the expression by dividing the numerator and denominator by their greatest common divisor.

Q: What is the difference between the solutions of a quadratic equation?

A: The solutions of a quadratic equation are the values of $x$ that satisfy the equation. The difference between the solutions is the distance between the two solutions on the number line.

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

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in function to solve quadratic equations. Simply enter the coefficients of the equation and the calculator will give you the solutions.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. Simply enter the equation and the graphing tool will display the graph of the equation.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph of the equation where the parabola changes direction. The vertex is the minimum or maximum point of the parabola.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you can use the following formula:

$x = -\frac{b}{2a}$

This formula gives you the x-coordinate of the vertex. To find the y-coordinate of the vertex, substitute the x-coordinate into the equation and solve for $y$.

Q: Can I use a quadratic equation to model real-world problems?

A: Yes, you can use a quadratic equation to model real-world problems. Quadratic equations are used in physics to describe the motion of objects under the influence of gravity, in engineering to design bridges and buildings, and in economics to model the behavior of markets.

Q: What are some common applications of quadratic equations?

A: Some common applications of quadratic equations include:

• Physics: Quadratic equations are used to describe 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 markets.
• Computer Science: Quadratic equations are used in computer graphics and game development.

Q: How do I choose the right method to solve a quadratic equation?

A: To choose the right method to solve a quadratic equation, consider the following factors:

• Difficulty of the equation: If the equation is simple, you can use the quadratic formula. If the equation is complex, you may need to use a different method.
• Availability of technology: If you have access to a calculator or computer program, you can use it to solve the equation.
• Time constraints: If you are short on time, you may need to use a quicker method to solve the equation.

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

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

• Dividing by zero: Make sure to check for zero denominators before dividing.
• Simplifying incorrectly: Make sure to simplify the expression correctly.
• Forgetting to check for extraneous solutions: Make sure to check for extraneous solutions, especially when using the quadratic formula.

Q: How do I check my work when solving quadratic equations?

A: To check your work when solving quadratic equations, consider the following steps:

• Plug in the solutions: Plug the solutions back into the original equation to make sure they are true.
• Check for extraneous solutions: Check for extraneous solutions, especially when using the quadratic formula.
• Simplify the expression: Simplify the expression to make sure it is correct.