What Value Of $b$ Makes The Trinomial Below A Perfect Square?$x^2 - Bx + 100$A. 20 B. 10 C. 5 D. 50

What Value of b Makes the Trinomial a Perfect Square?

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 the value of $b$ that makes the trinomial $x^2 - bx + 100$ a perfect square.

Understanding Perfect Square Trinomials

A perfect square trinomial can be written in the form $(x + a)^2$ or $(x - a)^2$. When we expand these expressions, we get:

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

Comparing these expressions with the trinomial $x^2 - bx + 100$, we can see that the middle term $-bx$ must be equal to $-2ax$ or $2ax$, depending on whether we are using $(x + a)^2$ or $(x - a)^2$. This means that $b$ must be equal to $2a$ or $-2a$.

Finding the Value of b

To find the value of $b$, we need to find the value of $a$ that makes the trinomial a perfect square. We can do this by comparing the constant term $100$ with the constant term $a^2$.

• If we are using $(x + a)^2$, then $a^2 = 100$, which means that $a = \pm 10$.
• If we are using $(x - a)^2$, then $a^2 = 100$, which means that $a = \pm 10$.

In both cases, we have $a = \pm 10$. Since $b = 2a$ or $-2a$, we have:

• $b = 2(\pm 10) = \pm 20$
• $b = -2(\pm 10) = \mp 20$

However, we are given four options for the value of $b$: 20, 10, 5, and 50. We can see that only one of these options is equal to $\pm 20$.

Based on our analysis, we can conclude that the value of $b$ that makes the trinomial $x^2 - bx + 100$ a perfect square is $\boxed{20}$.

The correct answer is A. 20.

It's worth noting that the value of $b$ that makes the trinomial a perfect square is not unique. We can also have $b = -20$, which would make the trinomial a perfect square as well. However, this is not one of the options given in the problem.

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 know if a trinomial is a perfect square?

A: To determine if a trinomial is a perfect square, 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 $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant.

Q: How do I find the value of b in a perfect square trinomial?

A: To find the value of $b$ in a perfect square trinomial, you need to find the value of $a$ that makes the trinomial a perfect square. You can do this by comparing the constant term with the constant term $a^2$.

Q: What is the relationship between b and a in a perfect square trinomial?

A: In a perfect square trinomial, the value of $b$ is equal to $2a$ or $-2a$, depending on whether we are using $(x + a)^2$ or $(x - a)^2$.

Q: Can a perfect square trinomial have multiple values of b?

A: Yes, a perfect square trinomial can have multiple values of $b$. For example, if we have a perfect square trinomial $(x + 10)^2$, then $b = 2(10) = 20$ or $b = -2(10) = -20$.

Q: How do I determine which value of b is correct?

A: To determine which value of $b$ is correct, you need to check if the trinomial can be factored into the square of a binomial using the value of $b$ you are considering.

Q: Can a perfect square trinomial have a negative value of b?

A: Yes, a perfect square trinomial can have a negative value of $b$. For example, if we have a perfect square trinomial $(x - 10)^2$, then $b = -2(10) = -20$.

Q: What is the significance of perfect square trinomials in algebra?

A: Perfect square trinomials are significant in algebra because they can be factored into the square of a binomial, which makes it easier to solve quadratic equations.

Q: Can perfect square trinomials be used to solve quadratic equations?

A: Yes, perfect square trinomials can be used to solve quadratic equations. By factoring the trinomial into the square of a binomial, we can solve for the variable $x$.

Q: What are some common applications of perfect square trinomials?

A: Perfect square trinomials have many common applications in algebra, including solving quadratic equations, factoring quadratic expressions, and graphing quadratic functions.

Q: Can perfect square trinomials be used in real-world applications?

A: Yes, perfect square trinomials can be used in real-world applications, such as physics, engineering, and economics. They are used to model and solve problems involving quadratic relationships.