Factor The Perfect Square Trinomial.${ J^2 + 40j + 400 }$Enter Your Answer In The Box.

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 of ${a^2 + 2ab + b^2}$ or ${a^2 - 2ab + b^2}$, where ${a}$ and ${b}$ are constants. In this article, we will focus on factoring the perfect square trinomial of the form ${a^2 + 2ab + b^2}$.

Why is Factoring Important?

Factoring is an essential skill in algebra that helps us simplify complex expressions and solve equations. By factoring a perfect square trinomial, we can rewrite it in a more compact and manageable form, making it easier to work with. This skill is also crucial in solving quadratic equations and inequalities.

The Formula for Factoring a Perfect Square Trinomial

The formula for factoring a perfect square trinomial is:

${a^2 + 2ab + b^2 = (a + b)^2}$

This formula states that a perfect square trinomial can be factored into the square of a binomial, where the binomial has the form of ${a + b}$.

Step-by-Step Guide to Factoring a Perfect Square Trinomial

Now that we have the formula, let's apply it to the given expression: ${j^2 + 40j + 400}$.

Step 1: Identify the Values of a and b

To factor the expression, we need to identify the values of ${a}$ and ${b}$. In this case, we can see that ${a = j}$ and ${b = 20}$, since ${2ab = 2(j)(20) = 40j}$.

Step 2: Apply the Formula

Now that we have the values of ${a}$ and ${b}$, we can apply the formula:

${j^2 + 40j + 400 = (j + 20)^2}$

Step 3: Simplify the Expression

The expression is now in the form of a perfect square trinomial, which can be simplified to:

${(j + 20)^2 = j^2 + 40j + 400}$

Conclusion

Factoring a perfect square trinomial is an essential skill in algebra that helps us simplify complex expressions and solve equations. By applying the formula and following the step-by-step guide, we can factor a perfect square trinomial into the square of a binomial. In this article, we have factored the expression ${j^2 + 40j + 400}$ using the formula and simplified it to ${(j + 20)^2}$.

Common Mistakes to Avoid

When factoring a perfect square trinomial, there are several common mistakes to avoid:

• Not identifying the values of a and b: Make sure to identify the values of ${a}$ and ${b}$ correctly, as this will affect the final result.
• Not applying the formula correctly: Double-check that you have applied the formula correctly, and that the expression is in the form of a perfect square trinomial.
• Not simplifying the expression: Make sure to simplify the expression to its simplest form, which is the square of a binomial.

Practice Problems

To practice factoring perfect square trinomials, try the following problems:

• Factor the expression ${x^2 + 14x + 49}$.
• Factor the expression ${y^2 - 12y + 36}$.
• Factor the expression ${z^2 + 16z + 64}$.

Real-World Applications

Factoring perfect square trinomials has several real-world applications, including:

• Simplifying complex expressions: Factoring perfect square trinomials can help simplify complex expressions and make them easier to work with.
• Solving quadratic equations: Factoring perfect square trinomials can help solve quadratic equations and inequalities.
• Optimizing problems: Factoring perfect square trinomials can help optimize problems and find the maximum or minimum value of a function.

Conclusion

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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 the form of ${a^2 + 2ab + b^2}$ or ${a^2 - 2ab + b^2}$, where ${a}$ and ${b}$ are constants.

Q: Why is factoring a perfect square trinomial important?

A: Factoring a perfect square trinomial is important because it helps us simplify complex expressions and solve equations. By factoring a perfect square trinomial, we can rewrite it in a more compact and manageable form, making it easier to work with.

Q: What is the formula for factoring a perfect square trinomial?

A: The formula for factoring a perfect square trinomial is:

${a^2 + 2ab + b^2 = (a + b)^2}$

This formula states that a perfect square trinomial can be factored into the square of a binomial, where the binomial has the form of ${a + b}$.

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

A: To identify the values of ${a}$ and ${b}$ in a perfect square trinomial, you need to look at the expression and find the values of ${a}$ and ${b}$ that satisfy the equation ${2ab = 2(a)(b)}$. In other words, you need to find the values of ${a}$ and ${b}$ that make the middle term of the expression equal to ${2ab}$.

Q: What if I have a perfect square trinomial with a negative sign in front of it?

A: If you have a perfect square trinomial with a negative sign in front of it, you can factor it using the formula:

${a^2 - 2ab + b^2 = (a - b)^2}$

This formula states that a perfect square trinomial with a negative sign in front of it can be factored into the square of a binomial, where the binomial has the form of ${a - b}$.

Q: Can I factor a perfect square trinomial with a coefficient in front of it?

A: Yes, you can factor a perfect square trinomial with a coefficient in front of it. To do this, you need to factor out the coefficient from the expression and then factor the remaining perfect square trinomial.

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 need to check if it has the form of ${a^2 + 2ab + b^2}$ or ${a^2 - 2ab + b^2}$. If it does, then it is a perfect square trinomial.

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

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

• Not identifying the values of ${a}$ and ${b}$ correctly
• Not applying the formula correctly
• Not simplifying the expression to its simplest form

Q: How can I practice factoring perfect square trinomials?

A: You can practice factoring perfect square trinomials by working through examples and exercises. You can also try factoring perfect square trinomials with different coefficients and variables.

Q: What are some real-world applications of factoring perfect square trinomials?

A: Some real-world applications of factoring perfect square trinomials include:

• Simplifying complex expressions
• Solving quadratic equations
• Optimizing problems

Conclusion

Factoring perfect square trinomials is an essential skill in algebra that helps us simplify complex expressions and solve equations. By applying the formula and following the step-by-step guide, we can factor a perfect square trinomial into the square of a binomial. In this article, we have answered some common questions about factoring perfect square trinomials and provided some tips and examples to help you practice.