Solve The Equation: { (4x + 1)(x - 8) = 0$}$

Introduction

In mathematics, equations are a fundamental concept that help us understand and describe various phenomena. Solving equations is a crucial skill that is essential in mathematics, science, and engineering. In this article, we will focus on solving a quadratic equation of the form $(4x + 1)(x - 8) = 0$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. It is in the form of a product of two binomials, where each binomial is a linear expression. The equation is $(4x + 1)(x - 8) = 0$. To solve this equation, we need to find the values of $x$ that make the equation true.

The Zero Product Property

The zero product property is a fundamental concept in algebra that states that if the product of two or more factors is zero, then at least one of the factors must be zero. In other words, if $(a)(b) = 0$, then either $a = 0$ or $b = 0$. We can apply this property to the given equation to find the solutions.

Solving the Equation

Using the zero product property, we can set each factor equal to zero and solve for $x$. Let's start with the first factor, $4x + 1 = 0$. To solve for $x$, we need to isolate $x$ on one side of the equation.

from sympy import symbols, Eq, solve
x = symbols('x')

eq1 = Eq(4*x + 1, 0)

solution1 = solve(eq1, x)
print(solution1)

The solution to the first equation is $x = -\frac{1}{4}$.

Now, let's move on to the second factor, $x - 8 = 0$. To solve for $x$, we need to isolate $x$ on one side of the equation.

# Define the equation
eq2 = Eq(x - 8, 0)

solution2 = solve(eq2, x)
print(solution2)

The solution to the second equation is $x = 8$.

Conclusion

In this article, we solved the quadratic equation $(4x + 1)(x - 8) = 0$ using the zero product property. We set each factor equal to zero and solved for $x$. The solutions to the equation are $x = -\frac{1}{4}$ and $x = 8$. These solutions satisfy the original equation, and we can verify this by plugging them back into the equation.

Tips and Tricks

• When solving quadratic equations, it's essential to use the zero product property to find the solutions.
• Make sure to isolate the variable on one side of the equation to solve for it.
• Use algebraic manipulations to simplify the equation and make it easier to solve.

Real-World Applications

Solving quadratic equations has numerous real-world applications in various fields, including:

• Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, such as supply and demand curves, and to make predictions about future economic trends.

Conclusion

Introduction

In our previous article, we solved the quadratic equation $(4x + 1)(x - 8) = 0$ using the zero product property. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the zero product property?

A: The zero product property is a fundamental concept in algebra that states that if the product of two or more factors is zero, then at least one of the factors must be zero. In other words, if $(a)(b) = 0$, then either $a = 0$ or $b = 0$.

Q: How do I apply the zero product property to solve a quadratic equation?

A: To apply the zero product property, you need to set each factor equal to zero and solve for the variable. For example, in the equation $(4x + 1)(x - 8) = 0$, you would set each factor equal to zero and solve for $x$.

Q: What are the solutions to the equation $(4x + 1)(x - 8) = 0$?

A: The solutions to the equation $(4x + 1)(x - 8) = 0$ are $x = -\frac{1}{4}$ and $x = 8$. These solutions satisfy the original equation, and you can verify this by plugging them back into the equation.

Q: How do I verify the solutions to a quadratic equation?

A: To verify the solutions to a quadratic equation, you need to plug the solutions back into the original equation and check if the equation is true. For example, you can plug $x = -\frac{1}{4}$ and $x = 8$ back into the equation $(4x + 1)(x - 8) = 0$ and check if the equation is true.

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

A: Solving quadratic equations has numerous real-world applications in various fields, including physics, engineering, and economics. For example, quadratic equations are used to describe the motion of objects under the influence of gravity, friction, and other forces.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to use algebraic manipulations to make the equation easier to solve. For example, you can use the distributive property to expand the equation and then combine like terms.

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

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

• Not using the zero product property to find the solutions.
• Not isolating the variable on one side of the equation.
• Not verifying the solutions by plugging them back into the original equation.

Conclusion

In conclusion, solving quadratic equations is a fundamental skill that is essential in mathematics, science, and engineering. By using the zero product property and algebraic manipulations, we can solve quadratic equations and find the solutions. We hope this Q&A guide has helped you understand the solution and answer any questions you may have.

Additional Resources

• Quadratic Equation Solver: A online tool that can help you solve quadratic equations.
• Algebraic Manipulations: A tutorial on algebraic manipulations that can help you simplify quadratic equations.
• Real-World Applications of Quadratic Equations: A tutorial on real-world applications of quadratic equations.

Final Tips

• Practice solving quadratic equations to become proficient in solving them.
• Use the zero product property to find the solutions to quadratic equations.
• Verify the solutions by plugging them back into the original equation.