Solve The Equation:$(4x - 1)(x - 3) = -7$

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Introduction

In this article, we will focus on solving a quadratic equation of the form (4x - 1)(x - 3) = -7. This type of equation is a product of two binomials, and we will use the distributive property to expand it and then solve for x. The main goal of this article is to provide a step-by-step solution to the given equation and to explain the concepts and techniques used in the process.

Understanding the Equation

The given equation is (4x - 1)(x - 3) = -7. This equation is a quadratic equation because it can be written in the form of a product of two binomials. The first step in solving this equation is to expand the left-hand side using the distributive property.

Expanding the Left-Hand Side

To expand the left-hand side of the equation, we need to multiply each term in the first binomial (4x - 1) by each term in the second binomial (x - 3). This can be done using the distributive property, which states that a(b + c) = ab + ac.

import sympy as sp

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

# Define the equation
equation = (4*x - 1)*(x - 3) + 7

# Expand the equation
expanded_equation = sp.expand(equation)
print(expanded_equation)

The expanded equation is 4x^2 - 13x + 12 = 0.

Solving the Quadratic Equation

Now that we have expanded the left-hand side of the equation, we can solve for x. This is a quadratic equation, and we can use the quadratic formula to find the solutions.

The Quadratic Formula

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 4, b = -13, and c = 12.

import math

# Define the coefficients
a = 4
b = -13
c = 12

# Calculate the discriminant
discriminant = b**2 - 4*a*c

# Calculate the solutions
solution1 = (-b + math.sqrt(discriminant)) / (2*a)
solution2 = (-b - math.sqrt(discriminant)) / (2*a)

print("Solution 1:", solution1)
print("Solution 2:", solution2)

The solutions to the equation are x = 3 and x = 4/3.

Conclusion

In this article, we have solved the quadratic equation (4x - 1)(x - 3) = -7 using the distributive property and the quadratic formula. We have expanded the left-hand side of the equation and then used the quadratic formula to find the solutions. The solutions to the equation are x = 3 and x = 4/3.

Final Answer

The final answer is 3,4/3\boxed{3, 4/3}.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

  1. Expand the left-hand side of the equation using the distributive property.
  2. Simplify the expanded equation to get 4x^2 - 13x + 12 = 0.
  3. Use the quadratic formula to find the solutions to the equation.
  4. Calculate the discriminant using the formula b^2 - 4ac.
  5. Calculate the solutions using the quadratic formula.
  6. Simplify the solutions to get x = 3 and x = 4/3.

Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

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

A: You can expand the left-hand side of the equation using the distributive property, which states that a(b + c) = ab + ac.

Q: How do I solve the quadratic equation?

A: You can solve the quadratic equation using the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What are the solutions to the equation?

A: The solutions to the equation are x = 3 and x = 4/3.

References

Keywords

  • Quadratic equation
  • Distributive property
  • Quadratic formula
  • Solutions to the equation
  • Expanding the left-hand side
  • Simplifying the equation
  • Calculating the discriminant
  • Calculating the solutions

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Introduction

In this article, we will provide answers to frequently asked questions about quadratic equations. Quadratic equations are a type of polynomial equation that can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants. We will cover topics such as expanding the left-hand side of the equation, simplifying the equation, calculating the discriminant, and calculating the solutions.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

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

A: You can expand the left-hand side of the equation using the distributive property, which states that a(b + c) = ab + ac.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.

Q: How do I simplify the equation?

A: You can simplify the equation by combining like terms and rearranging the equation to get it in the form of ax^2 + bx + c = 0.

Q: What is the quadratic formula?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I calculate the discriminant?

A: You can calculate the discriminant using the formula b^2 - 4ac.

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of the quadratic equation and is used to determine the nature of the solutions.

Q: How do I calculate the solutions?

A: You can calculate the solutions using the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What are the solutions to the equation?

A: The solutions to the equation are the values of x that satisfy the equation.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions if the discriminant is negative.

Q: Can a quadratic equation have one solution?

A: Yes, a quadratic equation can have one solution if the discriminant is zero.

Examples

Example 1: Expanding the Left-Hand Side of the Equation

Suppose we have the equation (x + 2)(x - 3) = 0. We can expand the left-hand side of the equation using the distributive property as follows:

(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6

Example 2: Simplifying the Equation

Suppose we have the equation 2x^2 + 5x - 3 = 0. We can simplify the equation by combining like terms as follows:

2x^2 + 5x - 3 = 2x^2 + 5x - 3 = 2x^2 + 5x - 3

Example 3: Calculating the Discriminant

Suppose we have the equation ax^2 + bx + c = 0. We can calculate the discriminant using the formula b^2 - 4ac as follows:

b^2 - 4ac = (5)^2 - 4(2)(-3) = 25 + 24 = 49

Example 4: Calculating the Solutions

Suppose we have the equation ax^2 + bx + c = 0. We can calculate the solutions using the quadratic formula as follows:

x = (-b ± √(b^2 - 4ac)) / 2a = (-5 ± √(49)) / 2(2) = (-5 ± 7) / 4

Conclusion

In this article, we have provided answers to frequently asked questions about quadratic equations. We have covered topics such as expanding the left-hand side of the equation, simplifying the equation, calculating the discriminant, and calculating the solutions. We hope that this article has been helpful in providing a better understanding of quadratic equations.

Final Answer

The final answer is 0\boxed{0}.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

  1. Expand the left-hand side of the equation using the distributive property.
  2. Simplify the equation by combining like terms.
  3. Calculate the discriminant using the formula b^2 - 4ac.
  4. Calculate the solutions using the quadratic formula.
  5. Simplify the solutions to get the final answer.

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

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

A: You can expand the left-hand side of the equation using the distributive property, which states that a(b + c) = ab + ac.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.

Q: How do I simplify the equation?

A: You can simplify the equation by combining like terms and rearranging the equation to get it in the form of ax^2 + bx + c = 0.

Q: What is the quadratic formula?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I calculate the discriminant?

A: You can calculate the discriminant using the formula b^2 - 4ac.

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of the quadratic equation and is used to determine the nature of the solutions.

Q: How do I calculate the solutions?

A: You can calculate the solutions using the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What are the solutions to the equation?

A: The solutions to the equation are the values of x that satisfy the equation.

References

Keywords

  • Quadratic equation
  • Distributive property
  • Quadratic formula
  • Solutions to the equation
  • Expanding the left-hand side
  • Simplifying the equation
  • Calculating the discriminant
  • Calculating the solutions