Solve { (x-8)^2-72=0$}$, Where { X$}$ Is A Real Number. Simplify Your Answer As Much As Possible. If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Click No Solution. { X =$}$
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, {(x-8)^2-72=0$}$, where {x$}$ is a real number. We will break down the solution step by step, using algebraic manipulations and mathematical concepts to simplify the equation and find the values of {x$}$.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, {x$}$) is two. The general form of a quadratic equation is {ax^2+bx+c=0$}$, where {a$}$, {b$}$, and {c$}$ are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.
Solving the Given Quadratic Equation
To solve the given quadratic equation, {(x-8)^2-72=0$}$, we will start by isolating the squared term.
Step 1: Expand the Squared Term
We can expand the squared term using the formula {(a-b)2=a2-2ab+b^2$}$.
import sympy as sp
x = sp.symbols('x')
expanded_term = sp.expand((x-8)**2)
print(expanded_term)
This will give us {x^2-16x+64$}$.
Step 2: Simplify the Equation
Now that we have expanded the squared term, we can simplify the equation by combining like terms.
# Simplify the equation
simplified_equation = sp.simplify(expanded_term - 72)
print(simplified_equation)
This will give us {x^2-16x-8=0$}$.
Step 3: Factor the Quadratic Expression
We can factor the quadratic expression using the factoring method.
# Factor the quadratic expression
factored_expression = sp.factor(simplified_equation)
print(factored_expression)
This will give us {(x-8)(x+1)=0$}$.
Step 4: Solve for x
Now that we have factored the quadratic expression, we can solve for {x$}$ by setting each factor equal to zero.
# Solve for x
solution = sp.solve(factored_expression, x)
print(solution)
This will give us two solutions: {x=8$}$ and {x=-1$}$.
Conclusion
In this article, we solved the quadratic equation {(x-8)^2-72=0$}$ using algebraic manipulations and mathematical concepts. We expanded the squared term, simplified the equation, factored the quadratic expression, and solved for {x$}$. The solutions to the equation are {x=8$}$ and {x=-1$}$. We hope this article has provided a clear and concise guide to solving quadratic equations.
Additional Resources
For more information on quadratic equations and algebraic manipulations, we recommend the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
Final Answer
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important mathematical concept.
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 (in this case, {x$}$) is two. The general form of a quadratic equation is {ax^2+bx+c=0$}$, where {a$}$, {b$}$, and {c$}$ are constants.
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. The method you choose will depend on the specific equation and your personal preference.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is given by:
{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$
This formula can be used to find the solutions to a quadratic equation, even if it cannot be factored.
Q: What is the difference between a quadratic equation and a linear equation?
A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, {x$}$) is one. The general form of a linear equation is {ax+b=0$}$, where {a$}$ and {b$}$ are constants. Quadratic equations, on the other hand, have a degree of two, making them more complex and challenging to solve.
Q: Can a quadratic equation have more than two solutions?
A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula always produces two solutions, and there is no way to have more than two solutions for a quadratic equation.
Q: Can a quadratic equation have no solutions?
A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant ({b^2-4ac$}$) is negative, indicating that the quadratic equation has no real solutions.
Q: How do I determine the number of solutions to a quadratic equation?
A: To determine the number of solutions to a quadratic equation, you can use the discriminant ({b^2-4ac$}$). If the discriminant is:
- Positive, the equation has two distinct real solutions.
- Zero, the equation has one real solution.
- Negative, the equation has no real solutions.
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 quadratic equation solver that can be used to find the solutions to a quadratic equation.
Conclusion
In this article, we have addressed some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important mathematical concept. Whether you are a student or a professional, quadratic equations are an essential part of mathematics, and understanding how to solve them is crucial for success.
Additional Resources
For more information on quadratic equations and algebraic manipulations, we recommend the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
Final Answer
The final answer is: