(4x-3)(5x-3) = (3x-1) (8x-7)

Mar 10, 2025 by ADMIN 29 views

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

Introduction

In algebra, equations are a fundamental concept that helps us solve for unknown variables. One of the most common types of equations is the quadratic equation, which is a polynomial equation of degree two. In this article, we will focus on solving a specific type of algebraic equation, namely the product of two binomials. We will use the given equation (4x-3)(5x-3) = (3x-1) (8x-7) as an example to demonstrate the step-by-step process of solving algebraic equations.

Understanding the Equation

Before we dive into solving the equation, let's first understand what it represents. The equation (4x-3)(5x-3) = (3x-1) (8x-7) is a product of two binomials, where each binomial is a polynomial expression with two terms. The left-hand side of the equation represents the product of two binomials, while the right-hand side represents the product of two other binomials.

Step 1: Expand the Left-Hand Side

To solve the equation, we need to expand the left-hand side of the equation. This involves multiplying each term in the first binomial by each term in the second binomial. We can use the distributive property to expand the left-hand side.

(4x-3)(5x-3) = 4x(5x-3) - 3(5x-3)

Using the distributive property, we can further expand the expression:

4x(5x-3) - 3(5x-3) = 20x^2 - 12x - 15x + 9

Combining like terms, we get:

20x^2 - 12x - 15x + 9 = 20x^2 - 27x + 9

Step 2: Expand the Right-Hand Side

Now that we have expanded the left-hand side of the equation, we need to expand the right-hand side. This involves multiplying each term in the first binomial by each term in the second binomial.

(3x-1)(8x-7) = 3x(8x-7) - 1(8x-7)

Using the distributive property, we can further expand the expression:

3x(8x-7) - 1(8x-7) = 24x^2 - 21x - 8x + 7

Combining like terms, we get:

24x^2 - 21x - 8x + 7 = 24x^2 - 29x + 7

Step 3: Equate the Expanded Expressions

Now that we have expanded both sides of the equation, we can equate the two expressions.

20x^2 - 27x + 9 = 24x^2 - 29x + 7

Step 4: Simplify the Equation

To simplify the equation, we can move all the terms to one side of the equation.

20x^2 - 27x + 9 - (24x^2 - 29x + 7) = 0

Using the distributive property, we can further simplify the expression:

20x^2 - 27x + 9 - 24x^2 + 29x - 7 = 0

Combining like terms, we get:

-4x^2 + 2x + 2 = 0

Step 5: Solve for x

Now that we have simplified the equation, we can solve for x. We can use the quadratic formula to solve for x.

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = -4, b = 2, and c = 2. Plugging these values into the quadratic formula, we get:

x = (-(2) ± √((2)^2 - 4(-4)(2))) / 2(-4)

Simplifying the expression, we get:

x = (-2 ± √(4 + 32)) / (-8)

Further simplifying the expression, we get:

x = (-2 ± √36) / (-8)

Simplifying the square root, we get:

x = (-2 ± 6) / (-8)

Simplifying the expression, we get two possible solutions for x:

x = (-2 + 6) / (-8) = 4/8 = 1/2
x = (-2 - 6) / (-8) = -8/8 = -1

Conclusion

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

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: There are several ways to solve a quadratic equation. One common method is to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula can be used to solve any quadratic equation of the form ax^2 + bx + c = 0.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by the equation: x = (-b ± √(b^2 - 4ac)) / 2a. This formula can be used to find the solutions to any quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression to find the solutions.

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

A: A real solution is a solution to a quadratic equation that is a real number. For example, if the quadratic equation x^2 + 4x + 4 = 0 has a solution of x = -2, then -2 is a real solution. An imaginary solution, on the other hand, is a solution to a quadratic equation that is a complex number. For example, if the quadratic equation x^2 + 4x + 4 = 0 has a solution of x = 2i, then 2i is an imaginary solution.

Q: How do I determine whether a quadratic equation has real or imaginary solutions?

A: To determine whether a quadratic equation has real or imaginary solutions, you can use the discriminant, which is given by the expression b^2 - 4ac. If the discriminant is positive, then the quadratic equation has two real solutions. If the discriminant is zero, then the quadratic equation has one real solution. If the discriminant is negative, then the quadratic equation has two imaginary solutions.

Q: What is the discriminant?

A: The discriminant is a mathematical expression that is used to determine the nature of the solutions to a quadratic equation. It is given by the expression b^2 - 4ac. If the discriminant is positive, then the quadratic equation has two real solutions. If the discriminant is zero, then the quadratic equation has one real solution. If the discriminant is negative, then the quadratic equation has two imaginary solutions.

Q: How do I use the discriminant to determine the nature of the solutions to a quadratic equation?

A: To use the discriminant to determine the nature of the solutions to a quadratic equation, you need to calculate the value of the discriminant. If the discriminant is positive, then the quadratic equation has two real solutions. If the discriminant is zero, then the quadratic equation has one real solution. If the discriminant is negative, then the quadratic equation has two imaginary solutions.

Q: What is the significance of the discriminant in quadratic equations?

A: The discriminant is a crucial concept in quadratic equations because it determines the nature of the solutions to the equation. If the discriminant is positive, then the quadratic equation has two real solutions, which can be used to solve the equation. If the discriminant is zero, then the quadratic equation has one real solution, which can be used to solve the equation. If the discriminant is negative, then the quadratic equation has two imaginary solutions, which cannot be used to solve the equation.

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

A: To apply the quadratic formula to solve a quadratic equation, you need to identify the values of a, b, and c in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

Not identifying the values of a, b, and c correctly
Not plugging the correct values into the quadratic formula
Not simplifying the expression correctly
Not checking the nature of the solutions (real or imaginary)

Q: How do I check the nature of the solutions to a quadratic equation?

A: To check the nature of the solutions to a quadratic equation, you can use the discriminant. If the discriminant is positive, then the quadratic equation has two real solutions. If the discriminant is zero, then the quadratic equation has one real solution. If the discriminant is negative, then the quadratic equation has two imaginary solutions.

Q: What are some real-world applications of the quadratic formula?

A: The quadratic formula has many real-world applications, including:

Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
Engineering: The quadratic formula is used to solve problems involving stress and strain in materials.
Economics: The quadratic formula is used to solve problems involving optimization, such as finding the maximum or minimum of a function.

Q: How do I use the quadratic formula to solve a system of linear equations?

A: To use the quadratic formula to solve a system of linear equations, you need to first convert the system of linear equations into a quadratic equation. Then, you can use the quadratic formula to solve the quadratic equation and find the solutions to the system of linear equations.