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Answer :
To make the trinomial [tex]\(x^2 - bx + 100\)[/tex] a perfect square, it should match the form [tex]\((x - a)^2\)[/tex]. Let’s break this down step-by-step:
1. Understanding the Perfect Square Formula:
A perfect square trinomial [tex]\((x - a)^2\)[/tex] expands to:
[tex]\[
x^2 - 2ax + a^2
\][/tex]
We want our trinomial [tex]\(x^2 - bx + 100\)[/tex] to match this form.
2. Identify the Square of a Constant:
Since the trinomial is [tex]\(x^2 - bx + 100\)[/tex], the constant term [tex]\(a^2\)[/tex] should be equal to 100.
3. Finding [tex]\(a\)[/tex]:
Solve [tex]\(a^2 = 100\)[/tex] to find [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{100} = 10
\][/tex]
4. Identifying the Coefficient [tex]\(b\)[/tex]:
In the perfect square form [tex]\((x - a)^2 = x^2 - 2ax + a^2\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-2a\)[/tex].
Given [tex]\(a = 10\)[/tex], calculate [tex]\(b\)[/tex]:
[tex]\[
b = 2 \times 10 = 20
\][/tex]
5. Conclusion:
The value of [tex]\(b\)[/tex] that makes the trinomial [tex]\(x^2 - bx + 100\)[/tex] a perfect square is 20.
Therefore, the correct answer is A. 20.
1. Understanding the Perfect Square Formula:
A perfect square trinomial [tex]\((x - a)^2\)[/tex] expands to:
[tex]\[
x^2 - 2ax + a^2
\][/tex]
We want our trinomial [tex]\(x^2 - bx + 100\)[/tex] to match this form.
2. Identify the Square of a Constant:
Since the trinomial is [tex]\(x^2 - bx + 100\)[/tex], the constant term [tex]\(a^2\)[/tex] should be equal to 100.
3. Finding [tex]\(a\)[/tex]:
Solve [tex]\(a^2 = 100\)[/tex] to find [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{100} = 10
\][/tex]
4. Identifying the Coefficient [tex]\(b\)[/tex]:
In the perfect square form [tex]\((x - a)^2 = x^2 - 2ax + a^2\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-2a\)[/tex].
Given [tex]\(a = 10\)[/tex], calculate [tex]\(b\)[/tex]:
[tex]\[
b = 2 \times 10 = 20
\][/tex]
5. Conclusion:
The value of [tex]\(b\)[/tex] that makes the trinomial [tex]\(x^2 - bx + 100\)[/tex] a perfect square is 20.
Therefore, the correct answer is A. 20.
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