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Solve the equation k^2-6k=11 by completing the square

Sagot :

Answer:

k = ±2√5 + 3

Step-by-step explanation:

  • k² - 6k + 9 = 11 + 9
  • (k - 3)² = 20
  • k - 3 = √20
  • k - 3 = ±2√5
  • k = ±2√5 + 3

Answer:

[tex]k=3+2\sqrt5\ or\ k=3-2\sqrt5[/tex]

Step-by-step explanation:

[tex]k^2-6k=11[/tex]

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Add one term in order to complete the square

[tex]k^2-6k+(6\times\frac{1}{2})^2=11+(6\times\frac{1}{2})^2[/tex]

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Calculate

[tex]k^2-6k+3^2=11+3^2[/tex]

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Factor the expression using [tex]a^2\pm 2ab+b^2=(a\pm b)^2[/tex]

[tex](k-3)^2=11+3^2[/tex]

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Calculate the power

[tex](k-3)^2=11+9[/tex]

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Calculate the sum or difference

[tex](k-3)^2=20[/tex]

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Split into two equations

[tex]k-3=\sqrt{20}\ or\ k-3=-\sqrt{20}[/tex]

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Move variables to the left side of the equation:

                                 [tex](k-3=\sqrt{20})[/tex]

[tex]k=\sqrt{20}+3\\ k=2\sqrt{5}+3[/tex]

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Move variables to the left side of the equation:

                               [tex](k-3=-\sqrt{20})[/tex]

[tex]k=\sqrt{20}+3\\ k=-2\sqrt{5}+3[/tex]

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So far:

[tex]k=2\sqrt{5}+3\ or\ k=-2\sqrt{5}+3[/tex]

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Find the union of the solutions

[tex]k=2\sqrt{5}+3\ or\ k=-2\sqrt{5}+3[/tex]

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I hope this helps you

:)

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