Answer :
To make the expression [tex]\(x^2 - 5x + c\)[/tex] a perfect square trinomial, it needs to fit the form [tex]\((x - a)^2\)[/tex]. A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It has the general form:
[tex]\[
(x - a)^2 = x^2 - 2ax + a^2
\][/tex]
We need to match our expression [tex]\(x^2 - 5x + c\)[/tex] with [tex]\(x^2 - 2ax + a^2\)[/tex].
1. Identify the term for [tex]\(-2ax\)[/tex]:
In the expression [tex]\(x^2 - 5x + c\)[/tex], the term [tex]\(-5x\)[/tex] corresponds to the [tex]\(-2ax\)[/tex] term in [tex]\((x - a)^2\)[/tex]. So, we have:
[tex]\[
-2a = -5
\][/tex]
2. Solve for [tex]\(a\)[/tex]:
We solve the equation for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{5}{2}
\][/tex]
3. Calculate [tex]\(c\)[/tex] using [tex]\(a^2\)[/tex]:
Now calculate [tex]\(c\)[/tex], which is [tex]\(a^2\)[/tex]. So, we have:
[tex]\[
c = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6.25
\][/tex]
Therefore, the value of [tex]\(c\)[/tex] that makes [tex]\(x^2 - 5x + c\)[/tex] a perfect square trinomial is [tex]\(6.25\)[/tex].
The correct answer is [tex]\((C) 6.25\)[/tex].
[tex]\[
(x - a)^2 = x^2 - 2ax + a^2
\][/tex]
We need to match our expression [tex]\(x^2 - 5x + c\)[/tex] with [tex]\(x^2 - 2ax + a^2\)[/tex].
1. Identify the term for [tex]\(-2ax\)[/tex]:
In the expression [tex]\(x^2 - 5x + c\)[/tex], the term [tex]\(-5x\)[/tex] corresponds to the [tex]\(-2ax\)[/tex] term in [tex]\((x - a)^2\)[/tex]. So, we have:
[tex]\[
-2a = -5
\][/tex]
2. Solve for [tex]\(a\)[/tex]:
We solve the equation for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{5}{2}
\][/tex]
3. Calculate [tex]\(c\)[/tex] using [tex]\(a^2\)[/tex]:
Now calculate [tex]\(c\)[/tex], which is [tex]\(a^2\)[/tex]. So, we have:
[tex]\[
c = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6.25
\][/tex]
Therefore, the value of [tex]\(c\)[/tex] that makes [tex]\(x^2 - 5x + c\)[/tex] a perfect square trinomial is [tex]\(6.25\)[/tex].
The correct answer is [tex]\((C) 6.25\)[/tex].
