Factorizing the Expression

Oct 13, 2025 by magicalceylontours.com 38 views

Introduction

In mathematics, factorizing and solving linear equations are fundamental concepts that form the basis of algebra. Factorizing involves expressing an algebraic expression as a product of simpler expressions, while solving linear equations involves finding the value of a variable that satisfies a given equation. In this article, we will focus on factorizing the expression $$cx + 15x$$ and then use the result to make $$x$$ the subject of the equation $$cx + 15x = q$$.

Factorizing the Expression

To factorize the expression $$cx + 15x$$, we need to identify the common factor. In this case, the common factor is $$x$$. We can factor out $$x$$ from both terms to get:

cx + 15x = x(c + 15)

This is the factorized form of the expression.

Understanding the Factorized Form

The factorized form of the expression $$cx + 15x$$ is $$x(c + 15)$$. This means that the original expression can be expressed as the product of two simpler expressions: $$x$$ and $$(c + 15)$$. The factor $$x$$ is common to both terms, and the factor $$(c + 15)$$ is the result of combining the remaining terms.

Using the Factorized Form to Make x the Subject

Now that we have factorized the expression $$cx + 15x$$, we can use the result to make $$x$$ the subject of the equation $$cx + 15x = q$$. To do this, we need to isolate $$x$$ on one side of the equation.

First, we can divide both sides of the equation by $$(c + 15)$$ to get:

\frac{cx + 15x}{c + 15} = \frac{q}{c + 15}

This simplifies to:

x = \frac{q}{c + 15}

Therefore, the value of $$x$$ that satisfies the equation $$cx + 15x = q$$ is $$x = \frac{q}{c + 15}$$.

Conclusion

In this article, we factorized the expression $$cx + 15x$$ and then used the result to make $$x$$ the subject of the equation $$cx + 15x = q$$. We showed that the factorized form of the expression is $$x(c + 15)$$ and then used this result to isolate $$x$$ on one side of the equation. The final answer is $$x = \frac{q}{c + 15}$$.

Example

Suppose we have the equation $$2x + 15x = 20$$. We can factorize the left-hand side of the equation as follows:

2x + 15x = x(2 + 15) = x(17)

Therefore, the equation becomes:

x(17) = 20

We can then divide both sides of the equation by 17 to get:

x = \frac{20}{17}

Therefore, the value of $$x$$ that satisfies the equation $$2x + 15x = 20$$ is $$x = \frac{20}{17}$$.

Tips and Tricks

When factorizing an expression, look for common factors.
When solving a linear equation, isolate the variable on one side of the equation.
When dividing both sides of an equation by a factor, make sure to divide both sides by the same factor.

Common Mistakes

Failing to identify common factors when factorizing an expression.
Failing to isolate the variable on one side of the equation when solving a linear equation.
Dividing both sides of an equation by the wrong factor.

Real-World Applications

Factorizing and solving linear equations have many real-world applications. For example, in physics, factorizing and solving linear equations are used to describe the motion of objects. In economics, factorizing and solving linear equations are used to model the behavior of economic systems. In computer science, factorizing and solving linear equations are used to optimize algorithms and solve problems.

Conclusion

Introduction

In our previous article, we discussed how to factorize the expression $$cx + 15x$$ and then use the result to make $$x$$ the subject of the equation $$cx + 15x = q$$. In this article, we will answer some frequently asked questions about factorizing and solving linear equations.

Q&A

Q: What is factorizing?

A: Factorizing is the process of expressing an algebraic expression as a product of simpler expressions.

Q: How do I factorize an expression?

A: To factorize an expression, look for common factors. A common factor is a factor that is common to all terms in the expression. For example, in the expression $$2x + 15x$$, the common factor is $$x$$.

Q: What is the difference between factorizing and simplifying?

A: Factorizing and simplifying are two different processes. Factorizing involves expressing an expression as a product of simpler expressions, while simplifying involves combining like terms to get a simpler expression.

Q: How do I solve a linear equation?

A: To solve a linear equation, isolate the variable on one side of the equation. This involves performing operations such as addition, subtraction, multiplication, and division to get the variable by itself.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I factorize a quadratic equation?

A: To factorize a quadratic equation, look for two numbers whose product is the constant term and whose sum is the coefficient of the variable. For example, in the quadratic equation $$x^2 + 5x + 6$$, the two numbers are 2 and 3, since 2 x 3 = 6 and 2 + 3 = 5.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation, while a system of linear equations is a set of two or more equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, use methods such as substitution or elimination to find the values of the variables that satisfy all the equations.

Q: What are some common mistakes to avoid when factorizing and solving linear equations?

A: Some common mistakes to avoid include failing to identify common factors, failing to isolate the variable on one side of the equation, and dividing both sides of an equation by the wrong factor.

Tips and Tricks

When factorizing an expression, look for common factors.
When solving a linear equation, isolate the variable on one side of the equation.
When dividing both sides of an equation by a factor, make sure to divide both sides by the same factor.
When solving a system of linear equations, use methods such as substitution or elimination to find the values of the variables that satisfy all the equations.

Real-World Applications

Conclusion

In conclusion, factorizing and solving linear equations are fundamental concepts in mathematics that have many real-world applications. By understanding how to factorize expressions and solve linear equations, we can solve a wide range of problems in physics, economics, computer science, and other fields.

Example Problems

Problem 1

Factorize the expression $$3x + 12x$$.

Solution

To factorize the expression $$3x + 12x$$, look for common factors. In this case, the common factor is $$x$$. Therefore, the factorized form of the expression is $$x(3 + 12) = x(15)$$.

Problem 2

Solve the linear equation $$2x + 5x = 20$$.

Solution

To solve the linear equation $$2x + 5x = 20$$, isolate the variable on one side of the equation. This involves performing operations such as addition, subtraction, multiplication, and division to get the variable by itself. In this case, we can add $$2x$$ and $$5x$$ to get $$7x$$, and then divide both sides of the equation by 7 to get $$x = \frac{20}{7}$$.

Problem 3

Factorize the quadratic equation $$x^2 + 5x + 6$$.

Solution

To factorize the quadratic equation $$x^2 + 5x + 6$$, look for two numbers whose product is the constant term and whose sum is the coefficient of the variable. In this case, the two numbers are 2 and 3, since 2 x 3 = 6 and 2 + 3 = 5. Therefore, the factorized form of the quadratic equation is $$(x + 2)(x + 3)$$.