Which Value Must Be Added To The Expression $x^2 + 12x$ To Make It A Perfect-square Trinomial?A. 6 B. 36 C. 72 D. 144

Introduction

In algebra, a perfect-square trinomial is a polynomial expression that can be factored into the square of a binomial. It is a crucial concept in mathematics, particularly in algebra and calculus. In this article, we will explore the concept of perfect-square trinomials and determine which value must be added to the expression $x^2 + 12x$ to make it a perfect-square trinomial.

What is a Perfect-Square Trinomial?

A perfect-square trinomial is a polynomial expression that can be written in the form $(a+b)^2$ or $(a-b)^2$. It is a quadratic expression that can be factored into the square of a binomial. For example, the expression $(x+3)^2$ is a perfect-square trinomial because it can be factored into $(x+3)(x+3)$.

The General Form of a Perfect-Square Trinomial

The general form of a perfect-square trinomial is $(a+b)^2 = a^2 + 2ab + b^2$ or $(a-b)^2 = a^2 - 2ab + b^2$. In this form, we can see that the expression is a quadratic expression with a leading coefficient of 1 and a constant term of $b^2$.

Determining the Value to be Added

To determine which value must be added to the expression $x^2 + 12x$ to make it a perfect-square trinomial, we need to find the value of $b^2$ in the general form of a perfect-square trinomial. We can do this by comparing the given expression with the general form of a perfect-square trinomial.

Step 1: Identify the Coefficient of the Linear Term

The coefficient of the linear term in the given expression is 12. This means that the value of $2ab$ in the general form of a perfect-square trinomial is 12.

Step 2: Determine the Value of $a^2$

Since the given expression is $x^2 + 12x$, we can see that the value of $a^2$ is $x^2$.

Step 3: Determine the Value of $b^2$

Now that we have the value of $a^2$, we can determine the value of $b^2$. We know that $2ab = 12$, so we can divide both sides of the equation by 2 to get $ab = 6$. Since $a^2 = x^2$, we can substitute this value into the equation $ab = 6$ to get $x^2b = 6$. Now, we can divide both sides of the equation by $x^2$ to get $b = \frac{6}{x^2}$. However, this is not a valid solution because $b$ must be a constant. Therefore, we need to re-examine our previous steps.

Re-examining the Previous Steps

Let's re-examine the previous steps to see if we can find a valid solution. In Step 1, we identified the coefficient of the linear term as 12. This means that the value of $2ab$ in the general form of a perfect-square trinomial is 12. We can rewrite this equation as $2ab = 12$. Now, we can divide both sides of the equation by 2 to get $ab = 6$. Since $a^2 = x^2$, we can substitute this value into the equation $ab = 6$ to get $x^2b = 6$. Now, we can divide both sides of the equation by $x^2$ to get $b = \frac{6}{x^2}$. However, this is not a valid solution because $b$ must be a constant.

A New Approach

Let's try a new approach. We know that the given expression is $x^2 + 12x$. We can rewrite this expression as $(x+6)^2 - 36$. This expression is a perfect-square trinomial because it can be factored into $(x+6)(x+6) - 36$.

Conclusion

In conclusion, the value that must be added to the expression $x^2 + 12x$ to make it a perfect-square trinomial is 36.

Final Answer

Q&A: Perfect-Square Trinomials

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 is a quadratic expression that can be written in the form $(a+b)^2$ or $(a-b)^2$.

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

A: The general form of a perfect-square trinomial is $(a+b)^2 = a^2 + 2ab + b^2$ or $(a-b)^2 = a^2 - 2ab + b^2$.

Q: How do I determine which value must be added to a given expression to make it a perfect-square trinomial?

A: To determine which value must be added to a given expression to make it a perfect-square trinomial, you need to find the value of $b^2$ in the general form of a perfect-square trinomial. You can do this by comparing the given expression with the general form of a perfect-square trinomial.

Q: What is the value that must be added to the expression $x^2 + 12x$ to make it a perfect-square trinomial?

A: The value that must be added to the expression $x^2 + 12x$ to make it a perfect-square trinomial is 36.

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

A: To factor a perfect-square trinomial, you need to find the values of $a$ and $b$ in the general form of a perfect-square trinomial. You can do this by comparing the given expression with the general form of a perfect-square trinomial.

Q: What is the difference between a perfect-square trinomial and a quadratic expression?

A: A perfect-square trinomial is a quadratic expression that can be factored into the square of a binomial. A quadratic expression is a polynomial expression of degree 2 that cannot be factored into the square of a binomial.

Q: Can a perfect-square trinomial have a negative value?

A: Yes, a perfect-square trinomial can have a negative value. For example, the expression $(x-3)^2$ is a perfect-square trinomial with a negative value.

Q: Can a perfect-square trinomial have a fractional value?

A: Yes, a perfect-square trinomial can have a fractional value. For example, the expression $(x+\frac{1}{2})^2$ is a perfect-square trinomial with a fractional value.

Q: Can a perfect-square trinomial have a complex value?

A: Yes, a perfect-square trinomial can have a complex value. For example, the expression $(x+i)^2$ is a perfect-square trinomial with a complex value.

Conclusion

In conclusion, perfect-square trinomials are an important concept in algebra that can be used to factor quadratic expressions. By understanding the general form of a perfect-square trinomial and how to determine which value must be added to a given expression to make it a perfect-square trinomial, you can solve a wide range of algebraic problems.

Final Answer

The final answer is 36.