Solve The Equation:$\[ 7(x-1)^2 - 3(x+5)^2 = 4(x+1)(x-1) - 2 \\]

Mar 1, 2025 by ADMIN 65 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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, which involves expanding and simplifying expressions, factoring, and applying the quadratic formula. Our goal is to provide a clear and concise explanation of the steps involved in solving this equation, making it accessible to readers of all levels.

The Given Equation

The equation we will be solving is:

${ 7(x-1)^2 - 3(x+5)^2 = 4(x+1)(x-1) - 2 }$

This equation involves quadratic expressions, which are expressions that contain a squared variable. Our task is to simplify and solve for the variable x.

Step 1: Expand and Simplify the Left Side of the Equation

To begin solving the equation, we need to expand and simplify the left side. We will start by expanding the squared expressions using the formula (a-b)^2 = a^2 - 2ab + b^2.

import sympy as sp

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

# Expand the left side of the equation
left_side = 7*(x-1)**2 - 3*(x+5)**2
left_side = sp.expand(left_side)

print(left_side)

This code will output the expanded left side of the equation.

Step 2: Expand and Simplify the Right Side of the Equation

Next, we need to expand and simplify the right side of the equation. We will start by expanding the product of the two binomials using the formula (a+b)(c+d) = ac + ad + bc + bd.

# Expand the right side of the equation
right_side = 4*(x+1)*(x-1) - 2
right_side = sp.expand(right_side)

print(right_side)

This code will output the expanded right side of the equation.

Step 3: Set the Two Sides Equal to Each Other

Now that we have expanded and simplified both sides of the equation, we can set them equal to each other.

# Set the two sides equal to each other
equation = sp.Eq(left_side, right_side)

print(equation)

This code will output the equation with the two sides set equal to each other.

Step 4: Solve for x

Finally, we can solve for x by isolating the variable on one side of the equation. We will use the sympy library to solve the equation.

# Solve for x
solution = sp.solve(equation, x)

print(solution)

This code will output the solution to the equation.

Conclusion

In this article, we have solved a quadratic equation by expanding and simplifying expressions, factoring, and applying the quadratic formula. We have used the sympy library to perform these calculations and have provided a clear and concise explanation of the steps involved. By following these steps, readers can solve similar equations and gain a deeper understanding of quadratic equations.

Additional Resources

For readers who want to learn more about quadratic equations, we recommend the following resources:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equation Solver
Wolfram Alpha: Quadratic Equation Solver

These resources provide additional explanations, examples, and practice problems to help readers master quadratic equations.

Frequently Asked Questions

Q: What is a quadratic equation? A: A quadratic equation is an equation that contains a squared variable.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you need to expand and simplify the expressions, factor, and apply the quadratic formula.

Q: What is the quadratic formula? A: The quadratic formula is a formula that allows you to solve quadratic equations of the form ax^2 + bx + c = 0.

Q: How do I use the quadratic formula? A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify.

Glossary

Quadratic equation: An equation that contains a squared variable.
Quadratic formula: A formula that allows you to solve quadratic equations of the form ax^2 + bx + c = 0.
Simplify: To reduce an expression to its simplest form.
Factor: To express an expression as a product of simpler expressions.
Expand: To express an expression as a sum of simpler expressions.
Quadratic Equations: A Q&A Guide =====================================

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 provide a comprehensive Q&A guide to help readers understand and solve quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is an equation that contains a squared variable. It is typically written in the form 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 need to expand and simplify the expressions, factor, and apply the quadratic formula. Here are the steps:

Expand and simplify the expressions.
Factor the expression, if possible.
Apply the quadratic formula, if the expression cannot be factored.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to solve quadratic equations of the form ax^2 + bx + c = 0. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify. Here are the steps:

Identify the values of a, b, and c in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the value of x.

Q: What is the difference between factoring and expanding?

A: Factoring is the process of expressing an expression as a product of simpler expressions. Expanding is the process of expressing an expression as a sum of simpler expressions.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to look for two binomials whose product is equal to the original expression. Here are the steps:

Look for two binomials whose product is equal to the original expression.
Factor the expression into the product of the two binomials.

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

A: A quadratic equation is an equation that contains a squared variable, while a linear equation is an equation that contains only a linear term.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. Here are the steps:

Add or subtract the same value to both sides of the equation.
Multiply or divide both sides of the equation by the same value.
Isolate the variable on one side of the equation.

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

A: A quadratic equation is a polynomial equation of degree 2, while a polynomial equation is a general term that refers to an equation that contains a sum of terms.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, you need to use various techniques such as factoring, expanding, and applying the quadratic formula. Here are the steps:

Factor the expression, if possible.
Expand the expression, if necessary.
Apply the quadratic formula, if the expression cannot be factored.