Which Value Must Be Added To The Expression $x^2 - 3x$ To Make It A Perfect-square Trinomial?A. $\frac{3}{2}$B. $\frac{9}{4}$C. 6D. 9

Introduction

In algebra, a perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It has a specific form: $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant. In this article, we will explore how to add a value to the expression $x^2 - 3x$ to make it a perfect-square trinomial.

What is a Perfect-Square Trinomial?

A perfect-square trinomial is a quadratic expression that can be written in the form $(x + a)^2$ or $(x - a)^2$. For example, $(x + 2)^2 = x^2 + 4x + 4$ and $(x - 3)^2 = x^2 - 6x + 9$. To identify a perfect-square trinomial, we need to look for the following characteristics:

• The expression has three terms: $x^2$, $bx$, and $c$.
• The coefficient of the $x^2$ term is 1.
• The coefficient of the $x$ term is twice the value of the constant term.
• The constant term is the square of the value of the constant term.

How to Identify a Perfect-Square Trinomial

To identify a perfect-square trinomial, we need to look for the characteristics mentioned above. We can also use the following formula to check if an expression is a perfect-square trinomial:

$(x + a)^2 = x^2 + 2ax + a^2$

If we can rewrite the expression in this form, then it is a perfect-square trinomial.

Adding a Value to the Expression $x^2 - 3x$

To add a value to the expression $x^2 - 3x$ to make it a perfect-square trinomial, we need to find the value of $a$ that makes the expression $(x + a)^2$. We can do this by comparing the expression $x^2 - 3x$ with the formula $(x + a)^2 = x^2 + 2ax + a^2$.

Step 1: Identify the Coefficient of the $x$ Term

The coefficient of the $x$ term in the expression $x^2 - 3x$ is -3. This means that the value of $2a$ is -3.

Step 2: Solve for $a$

To solve for $a$, we can divide both sides of the equation $2a = -3$ by 2.

$a = \frac{-3}{2}$

Step 3: Find the Value to Add

To find the value to add to the expression $x^2 - 3x$, we need to find the value of $a^2$. We can do this by squaring the value of $a$.

$a^2 = \left(\frac{-3}{2}\right)^2 = \frac{9}{4}$

Conclusion

In this article, we have explored how to add a value to the expression $x^2 - 3x$ to make it a perfect-square trinomial. We have identified the characteristics of a perfect-square trinomial and used the formula $(x + a)^2 = x^2 + 2ax + a^2$ to find the value of $a$ that makes the expression $(x + a)^2$. We have also found the value to add to the expression $x^2 - 3x$ by squaring the value of $a$. The value to add is $\frac{9}{4}$.

Answer

The value to add to the expression $x^2 - 3x$ to make it a perfect-square trinomial is $\frac{9}{4}$.

Final Answer

$$

Q&A: Perfect-Square Trinomials

Q: What is a perfect-square trinomial?

A: A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It has a specific form: $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant.

Q: How do I identify a perfect-square trinomial?

A: To identify a perfect-square trinomial, look for the following characteristics:

• The expression has three terms: $x^2$, $bx$, and $c$.
• The coefficient of the $x^2$ term is 1.
• The coefficient of the $x$ term is twice the value of the constant term.
• The constant term is the square of the value of the constant term.

Q: How do I check if an expression is a perfect-square trinomial?

A: You can use the following formula to check if an expression is a perfect-square trinomial:

$(x + a)^2 = x^2 + 2ax + a^2$

If you can rewrite the expression in this form, then it is a perfect-square trinomial.

Q: How do I add a value to an expression to make it a perfect-square trinomial?

A: To add a value to an expression to make it a perfect-square trinomial, follow these steps:

1. Identify the coefficient of the $x$ term in the expression.
2. Solve for $a$ by dividing the coefficient of the $x$ term by 2.
3. Find the value to add by squaring the value of $a$.

Q: What is the value to add to the expression $x^2 - 3x$ to make it a perfect-square trinomial?

A: The value to add to the expression $x^2 - 3x$ to make it a perfect-square trinomial is $\frac{9}{4}$.

Q: How do I factor a perfect-square trinomial?

A: To factor a perfect-square trinomial, use the following formula:

$(x + a)^2 = x^2 + 2ax + a^2$

If you can rewrite the expression in this form, then you can factor it as $(x + a)^2$.

Q: What are some examples of perfect-square trinomials?

A: Some examples of perfect-square trinomials include:

• $(x + 2)^2 = x^2 + 4x + 4$
• $(x - 3)^2 = x^2 - 6x + 9$
• $(x + 1)^2 = x^2 + 2x + 1$

Q: Can I use perfect-square trinomials to simplify expressions?

A: Yes, you can use perfect-square trinomials to simplify expressions. By factoring an expression as a perfect-square trinomial, you can rewrite it in a simpler form.

Conclusion

In this article, we have explored the concept of perfect-square trinomials and how to identify, add values to, and factor them. We have also provided examples and answers to common questions about perfect-square trinomials. By mastering perfect-square trinomials, you can simplify expressions and solve problems more efficiently.

Final Answer

The final answer is $\boxed{\frac{9}{4}}$.