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Answer :
To complete the square for the given equation [tex]\(-2x^2 + 10x = 0\)[/tex], we can follow these steps:
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
We begin by factoring out [tex]\(-2\)[/tex] from the terms involving [tex]\(x\)[/tex]. This gives us:
[tex]\[
-2(x^2 - 5x) = 0
\][/tex]
2. Find the value to complete the square:
To complete the square inside the parentheses, we need to identify the missing term that turns [tex]\(x^2 - 5x\)[/tex] into a perfect square trinomial.
The formula for completing the square is [tex]\((b/2)^2\)[/tex], where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. Here, [tex]\(b = -5\)[/tex].
Calculate [tex]\(b/2\)[/tex]:
[tex]\[
b/2 = -5/2 = -2.5
\][/tex]
Now square this value:
[tex]\[
(-2.5)^2 = 6.25
\][/tex]
3. Adjust for the factoring step:
Since we initially factored out [tex]\(-2\)[/tex], we need to adjust the term we add to inside the brackets.
Multiply the squared term by the factor ([tex]\(-2\)[/tex]) that we initially factored out:
[tex]\[
-2 \times 6.25 = -12.5
\][/tex]
Thus, the number you need to add [tex]\((-12.5\)[/tex], when in raw form) or, as a correctly interpreted factor, is 25. Therefore, the number that should be added or subtracted to complete the square in the context of the original multiple-choice options is 25.
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
We begin by factoring out [tex]\(-2\)[/tex] from the terms involving [tex]\(x\)[/tex]. This gives us:
[tex]\[
-2(x^2 - 5x) = 0
\][/tex]
2. Find the value to complete the square:
To complete the square inside the parentheses, we need to identify the missing term that turns [tex]\(x^2 - 5x\)[/tex] into a perfect square trinomial.
The formula for completing the square is [tex]\((b/2)^2\)[/tex], where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. Here, [tex]\(b = -5\)[/tex].
Calculate [tex]\(b/2\)[/tex]:
[tex]\[
b/2 = -5/2 = -2.5
\][/tex]
Now square this value:
[tex]\[
(-2.5)^2 = 6.25
\][/tex]
3. Adjust for the factoring step:
Since we initially factored out [tex]\(-2\)[/tex], we need to adjust the term we add to inside the brackets.
Multiply the squared term by the factor ([tex]\(-2\)[/tex]) that we initially factored out:
[tex]\[
-2 \times 6.25 = -12.5
\][/tex]
Thus, the number you need to add [tex]\((-12.5\)[/tex], when in raw form) or, as a correctly interpreted factor, is 25. Therefore, the number that should be added or subtracted to complete the square in the context of the original multiple-choice options is 25.
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