What Number Should Be Added To The Expression X 2 + 6 X X^2 + 6x X 2 + 6 X To Change It Into A Perfect Square Trinomial?A. 3 B. 6 C. 9 D. 12

Mar 12, 2025 by ADMIN 144 views

What Number Should Be Added to the Expression $x^2 + 6x$ to Change It into a Perfect Square Trinomial?

A perfect square trinomial is a polynomial expression that can be factored into the square of a binomial. It has a specific form, where the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms. In this article, we will explore how to change the expression $x^2 + 6x$ into a perfect square trinomial by adding a specific number.

Understanding Perfect Square Trinomials

A perfect square trinomial can be written in the form $(a+b)^2$ or $(a-b)^2$ , where $a$ and $b$ are constants. When we expand these expressions, we get $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ , respectively. To change the expression $x^2 + 6x$ into a perfect square trinomial, we need to find a number that, when added to the expression, will make it a perfect square trinomial.

Finding the Number to Add

To find the number to add, we need to identify the missing term in the perfect square trinomial. We know that the first term is $x^2$ and the last term is $6x$ . To make it a perfect square trinomial, we need to find a term that, when added to $x^2 + 6x$ , will make it a perfect square trinomial.

Let's assume that the perfect square trinomial is $(x+a)^2$ . When we expand this expression, we get $x^2 + 2ax + a^2$ . We can see that the middle term is $2ax$ , which is equal to $6x$ . Therefore, we can set up the equation $2ax = 6x$ .

Solving for $a$

To solve for $a$ , we can divide both sides of the equation by $2x$ . This gives us $a = \frac{6x}{2x} = 3$ . Therefore, the perfect square trinomial is $(x+3)^2$ .

Adding the Number to the Expression

Now that we have found the perfect square trinomial, we can add the number to the expression $x^2 + 6x$ . We can rewrite the expression as $(x+3)^2 - 9$ . When we expand this expression, we get $x^2 + 6x + 9 - 9 = x^2 + 6x$ .

However, we can also rewrite the expression as $(x+3)^2 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9 = x^2 + 6x + 9 - 9
Q&A: What Number Should Be Added to the Expression $x^2 + 6x$ to Change It into a Perfect Square Trinomial?

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a polynomial expression that can be factored into the square of a binomial. It has a specific form, where the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms.

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

A: To determine if an expression is a perfect square trinomial, you can try to factor it into the square of a binomial. If it can be factored in this way, then it is a perfect square trinomial.

Q: What is the general form of a perfect square trinomial?

A: The general form of a perfect square trinomial is $(a+b)^2$ or $(a-b)^2$ , where $a$ and $b$ are constants.

Q: How do I find the number to add to the expression $x^2 + 6x$ to change it into a perfect square trinomial?

A: To find the number to add, you need to identify the missing term in the perfect square trinomial. You can do this by setting up an equation and solving for the unknown term.

Q: What is the number to add to the expression $x^2 + 6x$ to change it into a perfect square trinomial?

A: The number to add is 9.

Q: Why is 9 the correct answer?

A: 9 is the correct answer because it is the value of the missing term in the perfect square trinomial $(x+3)^2$ . When we expand this expression, we get $x^2 + 6x + 9$ , which is equal to the original expression $x^2 + 6x$ plus 9.

Q: Can I use a different method to find the number to add?

A: Yes, you can use a different method to find the number to add. One way is to use the formula for the sum of squares, which is $a^2 + 2ab + b^2$ . You can set up an equation using this formula and solve for the unknown term.

Q: What are some common mistakes to avoid when working with perfect square trinomials?

A: Some common mistakes to avoid when working with perfect square trinomials include:

Not recognizing the general form of a perfect square trinomial
Not identifying the missing term in the perfect square trinomial
Not using the correct formula for the sum of squares
Not solving for the unknown term correctly

Q: How can I practice working with perfect square trinomials?

A: You can practice working with perfect square trinomials by trying out different examples and exercises. You can also use online resources or math textbooks to find additional practice problems.

Conclusion

In conclusion, the number to add to the expression $x^2 + 6x$ to change it into a perfect square trinomial is 9. This can be found by identifying the missing term in the perfect square trinomial and solving for the unknown term. By following the steps outlined in this article, you can master the concept of perfect square trinomials and become proficient in working with them.