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Chapter 5: Problem 85

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ m^{2}-14 m+c $$

Short Answer

Expert verified
The value of \( c \) is 49.

Step by step solution

01

Understanding a Perfect Square Trinomial

A perfect square trinomial is a polynomial that can be written in the form \( (a+b)^2 = a^2 + 2ab + b^2 \), or \( (a-b)^2 = a^2 - 2ab + b^2 \). The given trinomial is \( m^2 - 14m + c \). We need to find the value of \( c \) so that it fits this pattern.
02

Identifying the middle term

The middle term of the trinomial, \( -14m \), is compared with \(-2ab \) (from \( (a-b)^2 = a^2 - 2ab + b^2 \)). Here, \(-2ab = -14m \), which implies \( a = m \) and \( 2b = 14 \).
03

Solving for b

To solve for \( b \), we set \( 2b = 14 \). Dividing both sides by 2 gives \( b = 7 \).
04

Calculating the value of c

We know that \( c = b^2 \) for the trinomial to be a perfect square. Substituting \( b = 7 \) gives us \( c = 7^2 = 49 \).
05

Verifying the result

To verify, substitute \( c = 49 \) back into the original trinomial and check. The trinomial becomes \( m^2 - 14m + 49 \), which can be written as \( (m - 7)^2 \), confirming our solution that the trinomial is indeed a perfect square.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trinomial
A trinomial is a type of polynomial that consists of three distinct terms. It is expressed in the general form as \( ax^2 + bx + c \). In the given exercise, our trinomial is \( m^2 - 14m + c \), where:
  • \( a \) represents the coefficient of \( m^2 \), which is 1.
  • \( b \) is the coefficient of the middle term, \(-14\).
  • \( c \) is our constant term.
In this specific context, we are tasked with finding the value of \( c \) that transforms this trinomial into a perfect square trinomial, a specific type of trinomial that can be factored into a squared binomial.
Middle Term
The middle term of a polynomial plays a crucial role in identifying its properties and eventual transformation into a perfect square trinomial. In our example, the middle term is \(-14m\). This term is significant because it provides us with critical information about the other values in the trinomial.For a trinomial to be a perfect square, it should match the expanded form \((a-b)^2 = a^2 - 2ab + b^2\). The middle term equivalent \(-2ab\) in this pattern allows us to link the value of \(-14m\) to find the relationship between \( a \) and \( b \), knowing that in our example, \( a = m \) and thus establishing \(-2b = -14\).
Solving for b
To solve for \( b \), let's start with the equation derived from our middle term: \(-2b = -14\). This relationship comes from matching the middle term of the expanded binomial square. Solving for \( b \) involves straightforward algebraic manipulation:
  • First, you set up the equation: \( 2b = 14 \).
  • Divide both sides by 2: \( b = \frac{14}{2} \).
  • Calculate the division: \( b = 7 \).
Therefore, \( b \) is determined to be 7. With this value of \( b \), it will be possible to find the value of \( c \) that makes the trinomial a perfect square trinomial.
Verifying the Result
Verification is an essential step in confirming our calculations and ensuring that we have indeed transformed the trinomial into a perfect square trinomial. With the value \( b = 7 \), we find \( c \) using the equation \( c = b^2 \).
  • Substitute the value of \( b \): \( c = 7^2 \).
  • Calculate the square: \( c = 49 \).
Next, substitute \( c = 49 \) back into the original trinomial \( m^2 - 14m + c \) to confirm:
  • The expression becomes \( m^2 - 14m + 49 \).
  • This can be factored into \((m - 7)^2\).
This confirms our calculations, showing that our trinomial is indeed a perfect square. This process of verification helps ensure the accuracy of your mathematical transformations.

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Most popular questions from this chapter

Find the value of \(c\) that makes each trinomial a perfect square trinomial. Factor \(x^{6}-1\) completely, using the following methods from this chapter. A. Factor the expression by treating it as the difference of two squares, \(\left(x^{3}\right)^{2}-1^{2}\) B. Factor the expression by treating it as the difference of two squares, \(\left(x^{3}\right)^{2}-1^{2}\) C. Factor the expression by treating it as the difference of two squares, \(\left(x^{3}\right)^{2}-1^{2}\)

Multiply. See Section 5.4. $$ (x-2)\left(x^{2}+2 x+4\right) $$

Factor each trinomial. See Examples 5 through 10. $$ 6 x^{2}+11 x y+4 y^{2} $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}-x-12 $$

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(-5[P(x)]-Q(x)\)
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