Answer :
To determine which constant should be added to the polynomial [tex]\(x^2 - 10x\)[/tex] in order to form a perfect square trinomial, we need to follow these steps:
1. Identify the given polynomial:
The polynomial given is [tex]\(x^2 - 10x\)[/tex].
2. Recall the structure of a perfect square trinomial:
A perfect square trinomial has the form [tex]\(a^2 - 2ab + b^2 = (a - b)^2\)[/tex] (or similarly [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex]).
3. Determine the coefficient of [tex]\(x\)[/tex]:
In the given polynomial, the coefficient of [tex]\(x\)[/tex] is [tex]\(-10\)[/tex].
4. Find the value of [tex]\(b\)[/tex]:
In the term [tex]\( -10x \)[/tex], we can recognize it as [tex]\(-2ab\)[/tex]. Here [tex]\(a = x\)[/tex] and [tex]\( -2b \)[/tex] must equal [tex]\(-10\)[/tex].
Thus, we have:
[tex]\[ -2b = -10 \][/tex]
5. Solve for [tex]\(b\)[/tex]:
Dividing both sides by [tex]\(-2\)[/tex], we get:
[tex]\[ b = \frac{-10}{-2} = 5 \][/tex]
6. Calculate [tex]\(b^2\)[/tex]:
To complete the square, we need to add [tex]\(b^2\)[/tex]. Therefore:
[tex]\[ b = 5 \quad \Rightarrow \quad b^2 = 5^2 = 25 \][/tex]
So the constant that needs to be added to [tex]\(x^2 - 10x\)[/tex] to form a perfect square trinomial is [tex]\(25\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{25} \][/tex]
1. Identify the given polynomial:
The polynomial given is [tex]\(x^2 - 10x\)[/tex].
2. Recall the structure of a perfect square trinomial:
A perfect square trinomial has the form [tex]\(a^2 - 2ab + b^2 = (a - b)^2\)[/tex] (or similarly [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex]).
3. Determine the coefficient of [tex]\(x\)[/tex]:
In the given polynomial, the coefficient of [tex]\(x\)[/tex] is [tex]\(-10\)[/tex].
4. Find the value of [tex]\(b\)[/tex]:
In the term [tex]\( -10x \)[/tex], we can recognize it as [tex]\(-2ab\)[/tex]. Here [tex]\(a = x\)[/tex] and [tex]\( -2b \)[/tex] must equal [tex]\(-10\)[/tex].
Thus, we have:
[tex]\[ -2b = -10 \][/tex]
5. Solve for [tex]\(b\)[/tex]:
Dividing both sides by [tex]\(-2\)[/tex], we get:
[tex]\[ b = \frac{-10}{-2} = 5 \][/tex]
6. Calculate [tex]\(b^2\)[/tex]:
To complete the square, we need to add [tex]\(b^2\)[/tex]. Therefore:
[tex]\[ b = 5 \quad \Rightarrow \quad b^2 = 5^2 = 25 \][/tex]
So the constant that needs to be added to [tex]\(x^2 - 10x\)[/tex] to form a perfect square trinomial is [tex]\(25\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{25} \][/tex]
