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Answer :
To solve the equation [tex]\( x^2 - 10x + 27 = 0 \)[/tex] by completing the square, you can follow these steps:
1. Move the constant term to the other side:
Start with the equation:
[tex]\[
x^2 - 10x + 27 = 0
\][/tex]
Subtract 27 from both sides to isolate the quadratic and linear terms:
[tex]\[
x^2 - 10x = -27
\][/tex]
2. Complete the square:
To complete the square, take half of the coefficient of [tex]\( x \)[/tex] (which is -10), square it, and add it to both sides of the equation. The coefficient of [tex]\( x \)[/tex] is -10, half of -10 is -5, and squaring -5 gives 25.
Add 25 to both sides of the equation:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
3. Simplify both sides:
The left side of the equation is now a perfect square trinomial, and the right side simplifies to:
[tex]\[
x^2 - 10x + 25 = -2
\][/tex]
Therefore, the equation [tex]\( x^2 - 10x + 25 = -27 + 25 \)[/tex] is one of the steps Thomas could have taken to complete the square. This matches the calculation.
The correct equation showing a step in completing the square is:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
1. Move the constant term to the other side:
Start with the equation:
[tex]\[
x^2 - 10x + 27 = 0
\][/tex]
Subtract 27 from both sides to isolate the quadratic and linear terms:
[tex]\[
x^2 - 10x = -27
\][/tex]
2. Complete the square:
To complete the square, take half of the coefficient of [tex]\( x \)[/tex] (which is -10), square it, and add it to both sides of the equation. The coefficient of [tex]\( x \)[/tex] is -10, half of -10 is -5, and squaring -5 gives 25.
Add 25 to both sides of the equation:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
3. Simplify both sides:
The left side of the equation is now a perfect square trinomial, and the right side simplifies to:
[tex]\[
x^2 - 10x + 25 = -2
\][/tex]
Therefore, the equation [tex]\( x^2 - 10x + 25 = -27 + 25 \)[/tex] is one of the steps Thomas could have taken to complete the square. This matches the calculation.
The correct equation showing a step in completing the square is:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
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