Factor Completely: $4p^2 + 36p + 81$Express The Answer In The Form $(ap + B)^2$.Enter Your Answer In The Box:$\square$

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Understanding the Problem

To factor completely the given quadratic expression $4p^2 + 36p + 81$, we need to express it in the form $(ap + b)^2$. This involves identifying the values of $a$ and $b$ that satisfy the given expression. Factoring a quadratic expression in this form is a crucial skill in algebra, as it allows us to simplify complex expressions and solve equations.

Identifying the Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is $(ap + b)^2 = a^2p^2 + 2abp + b^2$. To factor the given expression, we need to identify the values of $a$ and $b$ that make it a perfect square trinomial.

Finding the Values of $a$ and $b$

To find the values of $a$ and $b$, we need to examine the given expression $4p^2 + 36p + 81$. We can start by looking at the first term, $4p^2$. This term is already in the form $a^2p^2$, where $a^2 = 4$. Taking the square root of both sides, we get $a = \pm 2$. Since the coefficient of the first term is positive, we take the positive value of $a$, which is $a = 2$.

Determining the Value of $b$

Now that we have found the value of $a$, we can determine the value of $b$. We can do this by examining the constant term, $81$. Since the constant term is equal to $b^2$, we can take the square root of both sides to get $b = \pm 9$. Since the coefficient of the linear term is positive, we take the positive value of $b$, which is $b = 9$.

Factoring the Expression

Now that we have found the values of $a$ and $b$, we can factor the expression $4p^2 + 36p + 81$ as $(2p + 9)^2$. This is a perfect square trinomial, as it can be written in the form $(ap + b)^2$.

Verifying the Factorization

To verify the factorization, we can expand the expression $(2p + 9)^2$ using the formula $(ap + b)^2 = a^2p^2 + 2abp + b^2$. Substituting $a = 2$ and $b = 9$, we get:

$(2p + 9)^2 = (2)^2p^2 + 2(2)(9)p + (9)^2$ $= 4p^2 + 36p + 81$

This shows that the factorization $(2p + 9)^2$ is correct, as it produces the original expression $4p^2 + 36p + 81$.

Conclusion

In this article, we have shown how to factor completely the quadratic expression $4p^2 + 36p + 81$. We identified the values of $a$ and $b$ that make the expression a perfect square trinomial, and then factored the expression as $(2p + 9)^2$. We also verified the factorization by expanding the expression and showing that it produces the original expression.

Tips and Tricks

• When factoring a quadratic expression, look for perfect square trinomials by examining the first and last terms.
• Use the formula $(ap + b)^2 = a^2p^2 + 2abp + b^2$ to expand the expression and verify the factorization.
• Make sure to take the positive value of $a$ and $b$ when factoring a quadratic expression.

Common Mistakes

• Failing to identify the perfect square trinomial.
• Not using the correct values of $a$ and $b$.
• Not verifying the factorization by expanding the expression.

Real-World Applications

Factoring quadratic expressions is a crucial skill in algebra, as it allows us to simplify complex expressions and solve equations. This skill is used in a variety of real-world applications, including:

• Physics: Factoring quadratic expressions is used to solve equations of motion and energy.
• Engineering: Factoring quadratic expressions is used to design and optimize systems.
• Computer Science: Factoring quadratic expressions is used in algorithms and data structures.

Final Answer

The final answer is: $\boxed{(2p + 9)^2}$

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Understanding the Problem

To factor completely the given quadratic expression $4p^2 + 36p + 81$, we need to express it in the form $(ap + b)^2$. This involves identifying the values of $a$ and $b$ that satisfy the given expression. Factoring a quadratic expression in this form is a crucial skill in algebra, as it allows us to simplify complex expressions and solve equations.

Q&A

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. The general form of a perfect square trinomial is $(ap + b)^2 = a^2p^2 + 2abp + b^2$.

Q: How do I identify the values of $a$ and $b$ in a perfect square trinomial?

A: To identify the values of $a$ and $b$, you need to examine the given expression. Look at the first term, $4p^2$. This term is already in the form $a^2p^2$, where $a^2 = 4$. Taking the square root of both sides, you get $a = \pm 2$. Since the coefficient of the first term is positive, you take the positive value of $a$, which is $a = 2$.

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

A: To determine the value of $b$, you need to examine the constant term, $81$. Since the constant term is equal to $b^2$, you can take the square root of both sides to get $b = \pm 9$. Since the coefficient of the linear term is positive, you take the positive value of $b$, which is $b = 9$.

Q: How do I factor a quadratic expression in the form $(ap + b)^2$?

A: To factor a quadratic expression in the form $(ap + b)^2$, you need to identify the values of $a$ and $b$ that satisfy the given expression. Then, you can write the expression as $(ap + b)^2$ and expand it using the formula $(ap + b)^2 = a^2p^2 + 2abp + b^2$.

Q: How do I verify the factorization of a quadratic expression?

A: To verify the factorization of a quadratic expression, you need to expand the expression using the formula $(ap + b)^2 = a^2p^2 + 2abp + b^2$. If the expanded expression matches the original expression, then the factorization is correct.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

A: Some common mistakes to avoid when factoring quadratic expressions include:

• Failing to identify the perfect square trinomial.
• Not using the correct values of $a$ and $b$.
• Not verifying the factorization by expanding the expression.

Q: What are some real-world applications of factoring quadratic expressions?

A: Factoring quadratic expressions is used in a variety of real-world applications, including:

• Physics: Factoring quadratic expressions is used to solve equations of motion and energy.
• Engineering: Factoring quadratic expressions is used to design and optimize systems.
• Computer Science: Factoring quadratic expressions is used in algorithms and data structures.

Tips and Tricks

• When factoring a quadratic expression, look for perfect square trinomials by examining the first and last terms.
• Use the formula $(ap + b)^2 = a^2p^2 + 2abp + b^2$ to expand the expression and verify the factorization.
• Make sure to take the positive value of $a$ and $b$ when factoring a quadratic expression.

Common Mistakes

• Failing to identify the perfect square trinomial.
• Not using the correct values of $a$ and $b$.
• Not verifying the factorization by expanding the expression.

Final Answer

The final answer is: $\boxed{(2p + 9)^2}$