Solve The Equation:$(x+7)^2 + 13(x+7) + 42 = 0$What Is The Solution Set? Select The Correct Choice Below And Fill In Any Answer Boxes In Your Choice.A. The Solution Set Is $\{\square\}$. (Use A Comma To Separate Answers As Needed.

Feb 27, 2025 by ADMIN 232 views

Solving the Quadratic Equation: $(x+7)^2 + 13(x+7) + 42 = 0$

In this article, we will focus on solving a quadratic equation of the form $(x+7)^2 + 13(x+7) + 42 = 0$ . This type of equation is a quadratic equation in disguise, and we will use various techniques to simplify and solve it. The solution set of this equation will be our main focus, and we will explore different methods to find the values of $x$ that satisfy the equation.

Understanding the Equation

The given equation is a quadratic equation in the form of $(x+7)^2 + 13(x+7) + 42 = 0$ . To solve this equation, we need to first simplify it by expanding the squared term and combining like terms.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the equation
equation = (x+7)**2 + 13*(x+7) + 42

# Expand the squared term
expanded_equation = sp.expand(equation)

# Print the expanded equation
print(expanded_equation)

Simplifying the Equation

After expanding the squared term, we get the equation $x^2 + 14x + 49 + 13x + 91 + 42 = 0$ . Combining like terms, we get $x^2 + 27x + 182 = 0$ .

# Define the simplified equation
simplified_equation = x**2 + 27*x + 182

# Print the simplified equation
print(simplified_equation)

Factoring the Equation

The simplified equation $x^2 + 27x + 182 = 0$ can be factored as $(x+13)(x+14) = 0$ .

# Define the factored equation
factored_equation = (x+13)*(x+14)

# Print the factored equation
print(factored_equation)

Solving for x

To solve for $x$ , we need to set each factor equal to zero and solve for $x$ . This gives us two possible solutions: $x+13=0$ and $x+14=0$ .

# Define the solutions
solution1 = sp.solve(x+13, x)
solution2 = sp.solve(x+14, x)

# Print the solutions
print(solution1)
print(solution2)

Conclusion

In conclusion, the solution set of the equation $(x+7)^2 + 13(x+7) + 42 = 0$ is $\{-13, -14\}$ . This means that the values of $x$ that satisfy the equation are $x=-13$ and $x=-14$ .

Final Answer

The final answer is: $\boxed{\{-13, -14\}}$
Solving the Quadratic Equation: $(x+7)^2 + 13(x+7) + 42 = 0$ - Q&A

In our previous article, we solved the quadratic equation $(x+7)^2 + 13(x+7) + 42 = 0$ and found the solution set to be $\{-13, -14\}$ . In this article, we will answer some frequently asked questions related to this equation and provide additional insights into solving quadratic equations.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable term. For example, $x^2 + 3x + 2 = 0$ is a quadratic equation. A linear equation, on the other hand, is a polynomial equation of degree one, which means it has only a variable term. For example, $2x + 3 = 0$ is a linear equation.

Q: How do I know if an equation is quadratic or linear?

A: To determine if an equation is quadratic or linear, look for the highest power of the variable. If the highest power is two, the equation is quadratic. If the highest power is one, the equation is linear.

Q: What are some common techniques for solving quadratic equations?

A: There are several techniques for solving quadratic equations, including:

Factoring: This involves expressing the quadratic equation as a product of two binomials.
Quadratic formula: This involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.
Graphing: This involves graphing the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula, and how do I use it?

A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . To use it, you need to identify the values of $a$ , $b$ , and $c$ in the quadratic equation. Then, plug these values into the formula and simplify.

Q: What is the difference between a real solution and a complex solution?

A: A real solution is a solution that can be expressed as a real number. For example, $x = 2$ is a real solution. A complex solution, on the other hand, is a solution that involves imaginary numbers. For example, $x = 2 + 3i$ is a complex solution.

Q: Can a quadratic equation have no real solutions?

A: Yes, a quadratic equation can have no real solutions. This occurs when the discriminant ( $b^2 - 4ac$ ) is negative.

Q: Can a quadratic equation have one real solution?

A: Yes, a quadratic equation can have one real solution. This occurs when the discriminant ( $b^2 - 4ac$ ) is zero.

Conclusion

In conclusion, solving quadratic equations can be a challenging task, but with the right techniques and tools, it can be done. We hope this Q&A article has provided you with a better understanding of quadratic equations and how to solve them.

Final Answer

The final answer is: $\boxed{\{-13, -14\}}$