1. Determine What Value Should Be Added To Each Of The Following Expressions To Make A Perfect Square Trinomial.a) $x^2 + 10x + 25$b) $x^2 + 14x + \square$Fill In The Square: $\square$

Introduction

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, which is $(a + b)^2 = a^2 + 2ab + b^2$. In this article, we will explore how to determine the value that needs to be added to each of the given expressions to make a perfect square trinomial.

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 $(a + b)^2 = a^2 + 2ab + b^2$. For example, $(x + 5)^2 = x^2 + 10x + 25$ is a perfect square trinomial.

Determining the Value to be Added

To determine the value that needs to be added to each of the given expressions to make a perfect square trinomial, we need to follow a specific procedure.

Step 1: Identify the Middle Term

The first step is to identify the middle term of the quadratic expression. The middle term is the term that is multiplied by the coefficient of the linear term.

Step 2: Determine the Value of the Binomial

Once we have identified the middle term, we need to determine the value of the binomial. The binomial is the expression that, when squared, will give us the quadratic expression.

Step 3: Square the Binomial

We need to square the binomial to get the quadratic expression. This will give us the value that needs to be added to the original expression.

Step 4: Add the Value to the Original Expression

Finally, we need to add the value that we obtained in the previous step to the original expression.

Example 1: $x^2 + 10x + 25$

Let's apply the steps to the first example.

Step 1: Identify the Middle Term

The middle term is $10x$.

Step 2: Determine the Value of the Binomial

The binomial is $(x + 5)$.

Step 3: Square the Binomial

$(x + 5)^2 = x^2 + 10x + 25$

Step 4: Add the Value to the Original Expression

Since the original expression is already a perfect square trinomial, we don't need to add any value.

Example 2: $x^2 + 14x + \square$

Let's apply the steps to the second example.

Step 1: Identify the Middle Term

The middle term is $14x$.

Step 2: Determine the Value of the Binomial

The binomial is $(x + 7)$.

Step 3: Square the Binomial

$(x + 7)^2 = x^2 + 14x + 49$

Step 4: Add the Value to the Original Expression

The value that needs to be added to the original expression is $49$.

Conclusion

In conclusion, determining the value that needs to be added to each of the given expressions to make a perfect square trinomial involves identifying the middle term, determining the value of the binomial, squaring the binomial, and adding the value to the original expression. By following these steps, we can easily determine the value that needs to be added to each of the given expressions to make a perfect square trinomial.

Q&A: Perfect Square Trinomials

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 $(a + b)^2 = a^2 + 2ab + b^2$.

Q: How do I determine the value that needs to be added to each of the given expressions to make a perfect square trinomial?

A: To determine the value that needs to be added to each of the given expressions to make a perfect square trinomial, you need to follow these steps:

1. Identify the middle term of the quadratic expression.
2. Determine the value of the binomial.
3. Square the binomial.
4. Add the value to the original expression.

Q: What is the middle term of a quadratic expression?

A: The middle term of a quadratic expression is the term that is multiplied by the coefficient of the linear term.

Q: How do I determine the value of the binomial?

A: To determine the value of the binomial, you need to look at the middle term and the constant term of the quadratic expression. The binomial is the expression that, when squared, will give you the quadratic expression.

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 of degree two, but it may not be a perfect square trinomial.

Q: Can a perfect square trinomial be factored into the product of two binomials?

A: Yes, a perfect square trinomial can be factored into the product of two binomials. This is because a perfect square trinomial has the form $(a + b)^2 = a^2 + 2ab + b^2$, which can be factored into $(a + b)(a + b)$.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to look at the middle term and the constant term of the quadratic expression. The binomial is the expression that, when squared, will give you the quadratic expression. You can then factor the quadratic expression into the product of two binomials.

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

• $(x + 5)^2 = x^2 + 10x + 25$
• $(x - 3)^2 = x^2 - 6x + 9$
• $(x + 2)^2 = x^2 + 4x + 4$

Q: Can a perfect square trinomial be used to solve a quadratic equation?

A: Yes, a perfect square trinomial can be used to solve a quadratic equation. If the quadratic equation is in the form of a perfect square trinomial, you can factor it into the product of two binomials and solve for the variable.

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

A: Perfect square trinomials have many real-world applications, including:

• Algebra: Perfect square trinomials are used to solve quadratic equations and to factor quadratic expressions.
• Geometry: Perfect square trinomials are used to find the area and perimeter of shapes.
• Physics: Perfect square trinomials are used to model the motion of objects.

Conclusion

In conclusion, perfect square trinomials are an important concept in algebra and have many real-world applications. By understanding how to determine the value that needs to be added to each of the given expressions to make a perfect square trinomial, you can solve quadratic equations and factor quadratic expressions.