What Value Of $c$ Makes $x^2 - 24x + C$ A Perfect Square Trinomial?A. -144 B. -48 C. 48 D. 144

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What Value of c Makes x^2 - 24x + 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 or (a - b)^2. In this article, we will explore the value of c that makes the quadratic expression x^2 - 24x + 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. It has the form (a + b)^2 or (a - b)^2, where a and b are constants. The general form of a perfect square trinomial is:

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

or

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

Expanding the Quadratic Expression

To determine the value of c that makes the quadratic expression x^2 - 24x + c a perfect square trinomial, we need to expand the expression and compare it with the general form of a perfect square trinomial.

x^2 - 24x + c = (x)^2 - 2(x)(12) + (12)^2

Expanding the expression, we get:

x^2 - 24x + c = x^2 - 24x + 144

Comparing with the General Form

Comparing the expanded expression with the general form of a perfect square trinomial, we can see that:

a^2 = x^2 2ab = -24x b^2 = 144

Solving for b

To solve for b, we can divide the second equation by 2a, which is 2x in this case.

2ab = -24x b = (-24x) / (2x) b = -12

Solving for c

Now that we have found the value of b, we can substitute it into the third equation to solve for c.

b^2 = 144 (-12)^2 = 144

This confirms that the value of b is indeed -12.

In conclusion, the value of c that makes the quadratic expression x^2 - 24x + c a perfect square trinomial is 144. This is because the expanded expression x^2 - 24x + c can be factored into the square of a binomial (x - 12)^2, which has the form (a - b)^2.

The correct answer is D. 144.

Here are a few additional examples of perfect square trinomials:

  • x^2 + 6x + 9 = (x + 3)^2
  • x^2 - 10x + 25 = (x - 5)^2
  • x^2 + 14x + 49 = (x + 7)^2

These examples demonstrate how to identify perfect square trinomials and how to factor them into the square of a binomial.

Here are a few tips and tricks for identifying perfect square trinomials:

  • Look for a quadratic expression that can be factored into the square of a binomial.
  • Check if the quadratic expression has the form (a + b)^2 or (a - b)^2.
  • Use the formula (a + b)^2 = a^2 + 2ab + b^2 to expand the expression and compare it with the general form of a perfect square trinomial.

Frequently Asked Questions

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 or (a - b)^2, where a and b are constants.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, look for a quadratic expression that can be factored into the square of a binomial. Check if the quadratic expression has the form (a + b)^2 or (a - b)^2.

Q: What is the general form of a perfect square trinomial?

A: The general form of a perfect square trinomial is:

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

or

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

Q: How do I expand a perfect square trinomial?

A: To expand a perfect square trinomial, use the formula (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2.

Q: What is the value of c in the quadratic expression x^2 - 24x + c that makes it a perfect square trinomial?

A: The value of c in the quadratic expression x^2 - 24x + c that makes it a perfect square trinomial is 144.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, use the formula (a + b)^2 = (a + b)(a + b) or (a - b)^2 = (a - b)(a - b).

Q: What are some examples of perfect square trinomials?

A: Here are a few examples of perfect square trinomials:

  • x^2 + 6x + 9 = (x + 3)^2
  • x^2 - 10x + 25 = (x - 5)^2
  • x^2 + 14x + 49 = (x + 7)^2

Q: How do I use perfect square trinomials in real-world applications?

A: Perfect square trinomials are used in a variety of real-world applications, including:

  • Algebra: Perfect square trinomials are used to factor quadratic expressions and solve equations.
  • 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.

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

A: Here are a few common mistakes to avoid when working with perfect square trinomials:

  • Not expanding the trinomial correctly
  • Not factoring the trinomial correctly
  • Not using the correct formula for the general form of a perfect square trinomial

In conclusion, perfect square trinomials are an important concept in algebra and are used in a variety of real-world applications. By understanding how to identify, expand, and factor perfect square trinomials, you can solve equations and model real-world situations with ease.

For more information on perfect square trinomials, check out the following resources:

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

By following these resources and practicing with perfect square trinomials, you can become proficient in this important algebra concept.