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Answer :
To make [tex]\(x^2 - 9x + c\)[/tex] a perfect square trinomial, it should match the form [tex]\((x - a)^2 = x^2 - 2ax + a^2\)[/tex].
Here are the step-by-step details to find the value of [tex]\(c\)[/tex]:
1. Identify the Coefficient of [tex]\(x\)[/tex]:
The given expression is [tex]\(x^2 - 9x + c\)[/tex]. The coefficient of [tex]\(x\)[/tex] is [tex]\(-9\)[/tex].
2. Relate it to the Perfect Square Form:
In a perfect square form [tex]\((x - a)^2\)[/tex], the middle term is [tex]\(-2ax\)[/tex]. So, we equate [tex]\(-2a\)[/tex] to the coefficient of [tex]\(x\)[/tex], which is [tex]\(-9\)[/tex]:
[tex]\[
-2a = -9
\][/tex]
3. Solve for [tex]\(a\)[/tex]:
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
a = \frac{9}{2}
\][/tex]
4. Find [tex]\(c\)[/tex]:
In the perfect square form, the constant term is [tex]\(a^2\)[/tex]. So, to make the expression a perfect square, [tex]\(c\)[/tex] should be:
[tex]\[
c = a^2 = \left(\frac{9}{2}\right)^2
\][/tex]
5. Calculate [tex]\(c\)[/tex]:
[tex]\[
c = \frac{81}{4}
\][/tex]
Therefore, the value of [tex]\(c\)[/tex] that makes [tex]\(x^2 - 9x + c\)[/tex] a perfect square trinomial is [tex]\(\frac{81}{4}\)[/tex]. Thus, the correct answer is option [tex]\( \text{F} \frac{81}{4} \)[/tex].
Here are the step-by-step details to find the value of [tex]\(c\)[/tex]:
1. Identify the Coefficient of [tex]\(x\)[/tex]:
The given expression is [tex]\(x^2 - 9x + c\)[/tex]. The coefficient of [tex]\(x\)[/tex] is [tex]\(-9\)[/tex].
2. Relate it to the Perfect Square Form:
In a perfect square form [tex]\((x - a)^2\)[/tex], the middle term is [tex]\(-2ax\)[/tex]. So, we equate [tex]\(-2a\)[/tex] to the coefficient of [tex]\(x\)[/tex], which is [tex]\(-9\)[/tex]:
[tex]\[
-2a = -9
\][/tex]
3. Solve for [tex]\(a\)[/tex]:
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
a = \frac{9}{2}
\][/tex]
4. Find [tex]\(c\)[/tex]:
In the perfect square form, the constant term is [tex]\(a^2\)[/tex]. So, to make the expression a perfect square, [tex]\(c\)[/tex] should be:
[tex]\[
c = a^2 = \left(\frac{9}{2}\right)^2
\][/tex]
5. Calculate [tex]\(c\)[/tex]:
[tex]\[
c = \frac{81}{4}
\][/tex]
Therefore, the value of [tex]\(c\)[/tex] that makes [tex]\(x^2 - 9x + c\)[/tex] a perfect square trinomial is [tex]\(\frac{81}{4}\)[/tex]. Thus, the correct answer is option [tex]\( \text{F} \frac{81}{4} \)[/tex].
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