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Answer :
To solve the equation [tex]\(x^2 - 10x + 27 = 0\)[/tex] by completing the square, Thomas would have taken the following steps:
1. Identify the Coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] in the equation is [tex]\(-10\)[/tex].
2. Calculate Half of the Coefficient:
Take half of [tex]\(-10\)[/tex], which is [tex]\(-5\)[/tex].
3. Square the Result from Step 2:
Square [tex]\(-5\)[/tex] to get [tex]\((-5)^2 = 25\)[/tex].
4. Complete the Square:
To complete the square, you add and subtract [tex]\(25\)[/tex] on the left side of the equation. This makes the quadratic expression a perfect square. The equation now becomes:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
5. Verify the Step:
The left side, [tex]\(x^2 - 10x + 25\)[/tex], can now be written as a perfect square trinomial:
[tex]\((x - 5)^2\)[/tex].
Thus, one of the steps Thomas could have taken to complete the square is represented by the equation:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
1. Identify the Coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] in the equation is [tex]\(-10\)[/tex].
2. Calculate Half of the Coefficient:
Take half of [tex]\(-10\)[/tex], which is [tex]\(-5\)[/tex].
3. Square the Result from Step 2:
Square [tex]\(-5\)[/tex] to get [tex]\((-5)^2 = 25\)[/tex].
4. Complete the Square:
To complete the square, you add and subtract [tex]\(25\)[/tex] on the left side of the equation. This makes the quadratic expression a perfect square. The equation now becomes:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
5. Verify the Step:
The left side, [tex]\(x^2 - 10x + 25\)[/tex], can now be written as a perfect square trinomial:
[tex]\((x - 5)^2\)[/tex].
Thus, one of the steps Thomas could have taken to complete the square is represented by the equation:
[tex]\[
x^2 - 10x + 25 = -27 + 25
\][/tex]
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