Continuing to Simplify the Equation

Oct 13, 2025 by talynt.org 123 views

Introduction

In algebra, we often come across expressions that have multiple factors. When we are given that a certain expression is a factor of another expression, it means that the first expression divides the second expression without leaving a remainder. In this article, we will explore how to find the value of $$a$$ in the given expression, given that $$3x + 4$$ is a factor of the expression.

Understanding the Problem

The given expression is $$12x^2 + ax - 20$$, and we are told that $$3x + 4$$ is a factor of this expression. This means that the expression can be divided by $$3x + 4$$ without leaving a remainder. To find the value of $$a$$, we need to use the fact that if $$3x + 4$$ is a factor of the expression, then the expression must be equal to zero when $$x$$ is equal to the value that makes $$3x + 4$$ equal to zero.

Finding the Value of $$x$$

To find the value of $$x$$ that makes $$3x + 4$$ equal to zero, we can set up the equation $$3x + 4 = 0$$ and solve for $$x$$. Subtracting 4 from both sides of the equation gives us $$3x = -4$$, and dividing both sides by 3 gives us $$x = -\frac{4}{3}$$.

Substituting the Value of $$x$$ into the Expression

Now that we have found the value of $$x$$, we can substitute it into the expression $$12x^2 + ax - 20$$ to get an equation in terms of $$a$$. Substituting $$x = -\frac{4}{3}$$ into the expression gives us:

12\left(-\frac{4}{3}\right)^2 + a\left(-\frac{4}{3}\right) - 20

Simplifying the Equation

To simplify the equation, we can start by evaluating the expression $$\left(-\frac{4}{3}\right)^2$$. This is equal to $$\frac{16}{9}$$, so the equation becomes:

12\left(\frac{16}{9}\right) + a\left(-\frac{4}{3}\right) - 20

Continuing to Simplify the Equation

Next, we can multiply 12 by $$\frac{16}{9}$$ to get $$\frac{192}{9}$$. The equation now becomes:

\frac{192}{9} + a\left(-\frac{4}{3}\right) - 20

Combining Like Terms

To combine like terms, we can start by getting rid of the fraction $$\frac{192}{9}$$ by multiplying both sides of the equation by 9. This gives us:

192 + 9a\left(-\frac{4}{3}\right) - 180

Continuing to Simplify the Equation

Next, we can simplify the term $$9a\left(-\frac{4}{3}\right)$$ by multiplying 9 by $$-\frac{4}{3}$$ to get $$-12a$$. The equation now becomes:

192 - 12a - 180

Combining Like Terms

To combine like terms, we can start by adding 192 and -180 to get 12. The equation now becomes:

12 - 12a

Solving for $$a$$

To solve for $$a$$, we can add 12a to both sides of the equation to get:

12 = 12a

Dividing Both Sides by 12

To solve for $$a$$, we can divide both sides of the equation by 12 to get:

a = \frac{12}{12}

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by 12 to get:

a = 1

Conclusion

In this article, we have shown how to find the value of $$a$$ in the given expression, given that $$3x + 4$$ is a factor of the expression. We started by finding the value of $$x$$ that makes $$3x + 4$$ equal to zero, and then substituted this value into the expression to get an equation in terms of $$a$$. We then simplified the equation and solved for $$a$$ to get $$a = 1$$.

Final Answer

The final answer is: $\boxed{1}$

Introduction

In our previous article, we explored how to find the value of $$a$$ in the given expression, given that $$3x + 4$$ is a factor of the expression. We used the fact that if $$3x + 4$$ is a factor of the expression, then the expression must be equal to zero when $$x$$ is equal to the value that makes $$3x + 4$$ equal to zero. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the significance of the value of $$x$$ that makes $$3x + 4$$ equal to zero?

A: The value of $$x$$ that makes $$3x + 4$$ equal to zero is significant because it is the value that makes the expression $$3x + 4$$ equal to zero. This value is used to substitute into the expression $$12x^2 + ax - 20$$ to get an equation in terms of $$a$$.

Q: How do we know that $$3x + 4$$ is a factor of the expression $$12x^2 + ax - 20$$?

A: We know that $$3x + 4$$ is a factor of the expression $$12x^2 + ax - 20$$ because the expression is equal to zero when $$x$$ is equal to the value that makes $$3x + 4$$ equal to zero.

Q: What is the process of finding the value of $$a$$ in the expression $$12x^2 + ax - 20$$?

A: The process of finding the value of $$a$$ in the expression $$12x^2 + ax - 20$$ involves the following steps:

Find the value of $$x$$ that makes $$3x + 4$$ equal to zero.
Substitute this value into the expression $$12x^2 + ax - 20$$ to get an equation in terms of $$a$$.
Simplify the equation and solve for $$a$$.

Q: What is the final answer for the value of $$a$$ in the expression $$12x^2 + ax - 20$$?

A: The final answer for the value of $$a$$ in the expression $$12x^2 + ax - 20$$ is $$a = 1$$.

Q: Can we use this method to find the value of $$a$$ in any expression?

A: No, this method can only be used to find the value of $$a$$ in an expression where $$3x + 4$$ is a factor of the expression.

Q: What are some common mistakes to avoid when finding the value of $$a$$ in an expression?

A: Some common mistakes to avoid when finding the value of $$a$$ in an expression include:

Not finding the correct value of $$x$$ that makes $$3x + 4$$ equal to zero.
Not substituting the correct value of $$x$$ into the expression.
Not simplifying the equation correctly.
Not solving for $$a$$ correctly.

Conclusion

In this article, we have answered some frequently asked questions related to finding the value of $$a$$ in the expression $$12x^2 + ax - 20$$. We have also highlighted some common mistakes to avoid when finding the value of $$a$$ in an expression. By following the steps outlined in this article, you should be able to find the value of $$a$$ in any expression where $$3x + 4$$ is a factor of the expression.

Final Answer

The final answer is: $\boxed{1}$