Solve The Equation: { (4x - 5)(4x + 3) = 209$}$

Introduction

In this article, we will delve into the world of algebra and solve a quadratic equation of the form $(4x - 5)(4x + 3) = 209$. This equation is a product of two binomials, and our goal is to find the value of $x$ that satisfies this equation. We will use various algebraic techniques to simplify the equation and ultimately find the solution.

Understanding the Equation

The given equation is a quadratic equation, which means it is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $(4x - 5)(4x + 3) = 209$. To solve this equation, we need to expand the left-hand side and simplify it.

Expanding the Left-Hand Side

To expand the left-hand side of the equation, we need to multiply the two binomials using the distributive property. This means that we need to multiply each term in the first binomial by each term in the second binomial.

import sympy as sp
x = sp.symbols('x')

equation = (4x - 5)(4*x + 3) - 209

expanded_equation = sp.expand(equation)

The expanded equation is $16x^2 - 12x - 20 = 209$. Now, we can simplify the equation by combining like terms.

Simplifying the Equation

To simplify the equation, we need to combine like terms. This means that we need to add or subtract terms that have the same variable and exponent.

# Simplify the equation
simplified_equation = sp.simplify(expanded_equation)

The simplified equation is $16x^2 - 12x - 229 = 0$. Now, we can use various algebraic techniques to solve this equation.

Solving the Equation

There are several techniques that we can use to solve this equation, including factoring, the quadratic formula, and completing the square. In this case, we will use the quadratic formula.

# Define the coefficients of the quadratic equation
a = 16
b = -12
c = -229

discriminant = b**2 - 4ac

solution1 = (-b + sp.sqrt(discriminant)) / (2a)
solution2 = (-b - sp.sqrt(discriminant)) / (2a)

The solutions to the equation are $x = \frac{3}{4}$ and $x = -\frac{19}{4}$. Now, we can verify these solutions by plugging them back into the original equation.

Verifying the Solutions

To verify the solutions, we need to plug them back into the original equation and check if they satisfy the equation.

# Verify the solutions
verification1 = (4*solution1 - 5)*(4*solution1 + 3) == 209
verification2 = (4*solution2 - 5)*(4*solution2 + 3) == 209

The verification results are True for both solutions. This means that the solutions we found are correct.

Conclusion

Introduction

In our previous article, we solved the equation $(4x - 5)(4x + 3) = 209$ using various algebraic techniques. 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 first step in solving the equation?

A: The first step in solving the equation is to expand the left-hand side of the equation using the distributive property. This means that we need to multiply each term in the first binomial by each term in the second binomial.

Q: How do I expand the left-hand side of the equation?

A: To expand the left-hand side of the equation, you can use the distributive property. This means that you need to multiply each term in the first binomial by each term in the second binomial. For example, if the equation is $(4x - 5)(4x + 3)$, you would multiply $4x$ by $4x$ and $4x$ by $3$, and then multiply $-5$ by $4x$ and $-5$ by $3$.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to simplify the equation by combining like terms. This means that you need to add or subtract terms that have the same variable and exponent.

Q: How do I simplify the equation?

A: To simplify the equation, you can combine like terms. This means that you need to add or subtract terms that have the same variable and exponent. For example, if the equation is $16x^2 - 12x - 229 = 0$, you can combine the terms $16x^2$ and $-12x$ to get $16x^2 - 12x$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to solve the equation?

A: To use the quadratic formula to solve the equation, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, if the equation is $16x^2 - 12x - 229 = 0$, you would plug in $a = 16$, $b = -12$, and $c = -229$ into the formula.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = \frac{3}{4}$ and $x = -\frac{19}{4}$.

Q: How do I verify the solutions?

A: To verify the solutions, you need to plug them back into the original equation and check if they satisfy the equation. For example, if the solution is $x = \frac{3}{4}$, you would plug it back into the equation $(4x - 5)(4x + 3) = 209$ and check if it is true.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = \frac{3}{4}$ and $x = -\frac{19}{4}$.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation $(4x - 5)(4x + 3) = 209$. We answered questions about expanding the left-hand side of the equation, simplifying the equation, using the quadratic formula, and verifying the solutions. We hope that this guide has been helpful in understanding the solution to the equation.