What Value Of $c$ Makes $x^2 + 6x + C$ A Perfect Square Trinomial?A. 3 B. 6 C. 9 D. 12

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What Value of c Makes x^2 + 6x + c a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. In other words, it is a quadratic expression that can be written in the form (a + b)^2, where a and b are constants. In this article, we will explore the value of c that makes the quadratic expression x^2 + 6x + c a perfect square trinomial.

Understanding Perfect Square Trinomials

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 (a + b)^2, where a and b are constants. When we expand the square of a binomial, we get a quadratic expression in the form a^2 + 2ab + b^2. Therefore, a perfect square trinomial can be written in the form a^2 + 2ab + b^2.

The Quadratic Expression x^2 + 6x + c

The quadratic expression x^2 + 6x + c is a perfect square trinomial if it can be factored into the square of a binomial. To determine the value of c that makes this expression a perfect square trinomial, we need to find the values of a and b such that a^2 + 2ab + b^2 = x^2 + 6x + c.

Comparing Coefficients

To compare the coefficients of the quadratic expression x^2 + 6x + c with the general form of a perfect square trinomial, we need to expand the square of a binomial. Let's assume that the binomial is (a + b). When we expand the square of this binomial, we get:

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

Comparing the coefficients of this expression with the quadratic expression x^2 + 6x + c, we get:

a^2 = x^2 2ab = 6x b^2 = c

Solving for a and b

To solve for a and b, we need to use the equations a^2 = x^2 and 2ab = 6x. From the first equation, we get a = x. Substituting this value of a into the second equation, we get:

2xb = 6x

Dividing both sides of this equation by 2x, we get:

b = 3

Finding the Value of c

Now that we have found the values of a and b, we can find the value of c. From the equation b^2 = c, we get:

c = b^2 = 3^2 = 9

In this article, we have explored the value of c that makes the quadratic expression x^2 + 6x + c a perfect square trinomial. We have compared the coefficients of the quadratic expression with the general form of a perfect square trinomial and solved for the values of a and b. Finally, we have found the value of c that makes the quadratic expression a perfect square trinomial.

The value of c that makes x^2 + 6x + c a perfect square trinomial is 9.

The final answer is 9.
Frequently Asked Questions (FAQs) About Perfect Square Trinomials

In our previous article, we explored the value of c that makes the quadratic expression x^2 + 6x + c a perfect square trinomial. In this article, we will answer some frequently asked questions (FAQs) about perfect square trinomials.

Q: What is a perfect square trinomial?

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. In other words, it is a quadratic expression that can be written in the form (a + b)^2, where a and b are constants.

Q: How do I determine if a quadratic expression is a perfect square trinomial?

To determine if a quadratic expression is a perfect square trinomial, you need to compare its coefficients with the general form of a perfect square trinomial. If the coefficients match, then the quadratic expression is a perfect square trinomial.

Q: What are the characteristics of a perfect square trinomial?

A perfect square trinomial has the following characteristics:

  • It can be factored into the square of a binomial.
  • It has a constant term that is a perfect square.
  • It has a coefficient of 1 on the x^2 term.

Q: How do I factor a perfect square trinomial?

To factor a perfect square trinomial, you need to find the values of a and b such that (a + b)^2 = x^2 + 6x + c. Once you have found the values of a and b, you can factor the perfect square trinomial as (a + b)^2.

Q: What is the relationship between a perfect square trinomial and a quadratic equation?

A perfect square trinomial is a special type of quadratic equation that can be factored into the square of a binomial. Quadratic equations that are not perfect square trinomials can be solved using other methods, such as the quadratic formula.

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

Yes, a perfect square trinomial can have a negative value. For example, the quadratic expression x^2 - 6x + 9 is a perfect square trinomial, and its value is negative when x is negative.

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

Yes, a perfect square trinomial can have a fractional value. For example, the quadratic expression x^2 + 2x + 1/4 is a perfect square trinomial, and its value is fractional when x is a fraction.

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

Yes, a perfect square trinomial can have a complex value. For example, the quadratic expression x^2 + 2x + i is a perfect square trinomial, and its value is complex when x is a complex number.

In this article, we have answered some frequently asked questions (FAQs) about perfect square trinomials. We hope that this article has provided you with a better understanding of perfect square trinomials and how to work with them.

If you want to learn more about perfect square trinomials, we recommend the following resources:

  • Khan Academy: Perfect Square Trinomials
  • Mathway: Perfect Square Trinomials
  • Wolfram Alpha: Perfect Square Trinomials

The final answer is 9.