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Answer :
To solve the equation [tex]\(x^2 - 10x + 27 = 0\)[/tex] by completing the square, Thomas can follow these steps:
1. Move the constant term to the other side of the equation:
Start with:
[tex]\[
x^2 - 10x + 27 = 0
\][/tex]
Subtract 27 from both sides:
[tex]\[
x^2 - 10x = -27
\][/tex]
2. Complete the square on the left side:
To complete the square, take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-10\)[/tex], and square it. Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex], and [tex]\((-5)^2 = 25\)[/tex].
3. Add and subtract this square term to/from the left side:
By adding and subtracting 25 on the left side, we complete the square:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
4. Rewrite the equation using the perfect square:
The left side now forms a perfect square trinomial:
[tex]\[
(x - 5)^2 = -2
\][/tex]
Thus, the step Thomas could have taken to complete the square is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
The correct choice from the options given is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
1. Move the constant term to the other side of the equation:
Start with:
[tex]\[
x^2 - 10x + 27 = 0
\][/tex]
Subtract 27 from both sides:
[tex]\[
x^2 - 10x = -27
\][/tex]
2. Complete the square on the left side:
To complete the square, take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-10\)[/tex], and square it. Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex], and [tex]\((-5)^2 = 25\)[/tex].
3. Add and subtract this square term to/from the left side:
By adding and subtracting 25 on the left side, we complete the square:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
4. Rewrite the equation using the perfect square:
The left side now forms a perfect square trinomial:
[tex]\[
(x - 5)^2 = -2
\][/tex]
Thus, the step Thomas could have taken to complete the square is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
The correct choice from the options given is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
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