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Answer :
To solve the equation [tex]\(x^2 - 10x + 27 = 0\)[/tex] by completing the square, I will guide you through the process step-by-step.
1. Start with the original equation:
[tex]\[ x^2 - 10x + 27 = 0 \][/tex]
2. Focus on the quadratic term and linear term:
We need to complete the square for the expression [tex]\(x^2 - 10x\)[/tex].
3. Find half of the coefficient of [tex]\(x\)[/tex] and square it:
The coefficient of [tex]\(x\)[/tex] is [tex]\(-10\)[/tex]. Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex]. Squaring [tex]\(-5\)[/tex] gives [tex]\(25\)[/tex]. This [tex]\(25\)[/tex] is the number we will use to complete the square.
4. Rewrite the equation by adding and subtracting this square inside the equation:
[tex]\[ x^2 - 10x + 25 - 25 + 27 = 0 \][/tex]
5. Simplify to complete the square within the equation:
[tex]\[ (x^2 - 10x + 25) + (27 - 25) = 0 \][/tex]
6. Rewrite the trinomial as a square:
[tex]\[ (x - 5)^2 + 2 = 0 \][/tex]
7. Rearrange the equation:
[tex]\[ (x - 5)^2 = -2 \][/tex]
Based on the steps we used, one of the key intermediate equations Thomas would have used to complete the square is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
So, the correct option is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
1. Start with the original equation:
[tex]\[ x^2 - 10x + 27 = 0 \][/tex]
2. Focus on the quadratic term and linear term:
We need to complete the square for the expression [tex]\(x^2 - 10x\)[/tex].
3. Find half of the coefficient of [tex]\(x\)[/tex] and square it:
The coefficient of [tex]\(x\)[/tex] is [tex]\(-10\)[/tex]. Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex]. Squaring [tex]\(-5\)[/tex] gives [tex]\(25\)[/tex]. This [tex]\(25\)[/tex] is the number we will use to complete the square.
4. Rewrite the equation by adding and subtracting this square inside the equation:
[tex]\[ x^2 - 10x + 25 - 25 + 27 = 0 \][/tex]
5. Simplify to complete the square within the equation:
[tex]\[ (x^2 - 10x + 25) + (27 - 25) = 0 \][/tex]
6. Rewrite the trinomial as a square:
[tex]\[ (x - 5)^2 + 2 = 0 \][/tex]
7. Rearrange the equation:
[tex]\[ (x - 5)^2 = -2 \][/tex]
Based on the steps we used, one of the key intermediate equations Thomas would have used to complete the square is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
So, the correct option is:
[tex]\[ x^2 - 10x + 25 = -27 + 25 \][/tex]
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