Complete The Expression So It Forms A Perfect-square Trinomial. X 2 − 5 X + □ X^2 - 5x + \square X 2 − 5 X + □

Mar 13, 2025 by ADMIN 111 views

**Complete the Expression to Form a Perfect-Square Trinomial**

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Introduction

In algebra, a perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form $(a + b)^2$ or $(a - b)^2$ , where $a$ and $b$ are constants. In this article, we will focus on completing the expression $x^2 - 5x + \square$ to form a perfect-square trinomial.

Understanding Perfect-Square Trinomials

A perfect-square trinomial is a quadratic expression that can be written in the form $(a + b)^2$ or $(a - b)^2$ . It has the following properties:

The first term is the square of the binomial.
The last term is the square of the binomial multiplied by the coefficient of the middle term.
The middle term is twice the product of the binomial and the coefficient of the last term.

For example, consider the perfect-square trinomial $(x + 3)^2$ . It can be expanded as follows:

$(x + 3)^2 = x^2 + 6x + 9$

In this example, the first term is $x^2$ , the middle term is $6x$ , and the last term is $9$ . We can see that the middle term is twice the product of $x$ and $3$ , which is $6x$ .

Completing the Expression

To complete the expression $x^2 - 5x + \square$ to form a perfect-square trinomial, we need to find the value of the last term. Let's call the last term $k$ . Then, we can write the expression as:

$x^2 - 5x + k$

We know that the middle term is $-5x$ , which is twice the product of $x$ and the coefficient of the last term. Let's call the coefficient of the last term $c$ . Then, we can write the middle term as:

$-5x = 2xc$

Simplifying the equation, we get:

$-5 = 2c$

Dividing both sides by $2$ , we get:

$c = -\frac{5}{2}$

Now, we can find the value of the last term $k$ . We know that the last term is the square of the binomial multiplied by the coefficient of the middle term. Let's call the binomial $a + b$ . Then, we can write the last term as:

$k = (a + b)^2 \cdot c$

Substituting the values of $a$ , $b$ , and $c$ , we get:

$k = (x - \frac{5}{2})^2 \cdot (-\frac{5}{2})$

Simplifying the equation, we get:

$k = \frac{25}{4}$

Therefore, the completed expression is:

$x^2 - 5x + \frac{25}{4}$

Verifying the Result

To verify the result, we can expand the completed expression as follows:

$x^2 - 5x + \frac{25}{4} = (x - \frac{5}{2})^2$

Expanding the binomial, we get:

$(x - \frac{5}{2})^2 = x^2 - 5x + \frac{25}{4}$

We can see that the completed expression is indeed a perfect-square trinomial.

Conclusion

In this article, we have learned how to complete the expression $x^2 - 5x + \square$ to form a perfect-square trinomial. We have also verified the result by expanding the completed expression. The completed expression is $x^2 - 5x + \frac{25}{4}$ , which is a perfect-square trinomial.

Examples and Exercises

Here are some examples and exercises to practice completing the expression to form a perfect-square trinomial:

Example 1

Complete the expression $x^2 + 6x + \square$ to form a perfect-square trinomial.

Solution

To complete the expression, we need to find the value of the last term. Let's call the last term $k$ . Then, we can write the expression as:

$x^2 + 6x + k$

We know that the middle term is $6x$ , which is twice the product of $x$ and the coefficient of the last term. Let's call the coefficient of the last term $c$ . Then, we can write the middle term as:

$6x = 2xc$

Simplifying the equation, we get:

$6 = 2c$

Dividing both sides by $2$ , we get:

$c = 3$

$k = (a + b)^2 \cdot c$

Substituting the values of $a$ , $b$ , and $c$ , we get:

$k = (x + 3)^2 \cdot 3$

Simplifying the equation, we get:

$k = 9$

Therefore, the completed expression is:

$x^2 + 6x + 9$

Exercise 1

Complete the expression $x^2 - 2x + \square$ to form a perfect-square trinomial.

Exercise 2

Complete the expression $x^2 + 4x + \square$ to form a perfect-square trinomial.

Exercise 3

Complete the expression $x^2 - 8x + \square$ to form a perfect-square trinomial.

Tips and Tricks

Here are some tips and tricks to help you complete the expression to form a perfect-square trinomial:

Make sure to identify the middle term and the coefficient of the last term.
Use the formula $k = (a + b)^2 \cdot c$ to find the value of the last term.
Simplify the equation to find the value of the last term.
Verify the result by expanding the completed expression.

By following these tips and tricks, you can easily complete the expression to form a perfect-square trinomial.

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Q1: What is a perfect-square trinomial?

A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form $(a + b)^2$ or $(a - b)^2$ , where $a$ and $b$ are constants.

Q2: How do I identify a perfect-square trinomial?

To identify a perfect-square trinomial, you need to check if the first term is the square of the binomial, the last term is the square of the binomial multiplied by the coefficient of the middle term, and the middle term is twice the product of the binomial and the coefficient of the last term.

Q3: How do I complete the expression to form a perfect-square trinomial?

To complete the expression to form a perfect-square trinomial, you need to follow these steps:

Identify the middle term and the coefficient of the last term.
Use the formula $k = (a + b)^2 \cdot c$ to find the value of the last term.
Simplify the equation to find the value of the last term.
Verify the result by expanding the completed expression.

Q4: What is the formula for completing the expression to form a perfect-square trinomial?

The formula for completing the expression to form a perfect-square trinomial is:

$k = (a + b)^2 \cdot c$

where $k$ is the value of the last term, $a$ and $b$ are the constants in the binomial, and $c$ is the coefficient of the middle term.

Q5: How do I verify the result after completing the expression to form a perfect-square trinomial?

To verify the result, you need to expand the completed expression and check if it matches the original expression.

Q6: What are some common mistakes to avoid when completing the expression to form a perfect-square trinomial?

Some common mistakes to avoid when completing the expression to form a perfect-square trinomial include:

Not identifying the middle term and the coefficient of the last term correctly.
Not using the correct formula to find the value of the last term.
Not simplifying the equation correctly.
Not verifying the result correctly.

Q7: Can I use the completing the square method to form a perfect-square trinomial?

Yes, you can use the completing the square method to form a perfect-square trinomial. The completing the square method involves adding and subtracting a constant term to create a perfect-square trinomial.

Q8: How do I use the completing the square method to form a perfect-square trinomial?

To use the completing the square method, you need to follow these steps:

Identify the middle term and the coefficient of the last term.
Add and subtract a constant term to create a perfect-square trinomial.
Simplify the equation to find the value of the last term.
Verify the result by expanding the completed expression.

Q9: What are some real-world applications of completing the expression to form a perfect-square trinomial?

Some real-world applications of completing the expression to form a perfect-square trinomial include:

Solving quadratic equations.
Finding the maximum or minimum value of a quadratic function.
Modeling real-world situations using quadratic functions.

Q10: Can I use a calculator to complete the expression to form a perfect-square trinomial?

Yes, you can use a calculator to complete the expression to form a perfect-square trinomial. However, it's always a good idea to verify the result by expanding the completed expression.

By following these FAQs, you can gain a better understanding of completing the expression to form a perfect-square trinomial and how to apply it in real-world situations.