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Solve by completing the square.\newlinek2−2k=15k^2 - 2k = 15\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____

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Q. Solve by completing the square.\newlinek2−2k=15k^2 - 2k = 15\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____
  1. Rewrite Equation: Write the equation in the form of completing the square.\newlineTo complete the square, the equation needs to be in the form k2−2k+(blank)=15+(blank)k^2 - 2k + (\text{blank}) = 15 + (\text{blank}).
  2. Calculate Completion Number: Calculate the number to complete the square.\newlineThe number needed to complete the square is given by (b2)2(\frac{b}{2})^2, where bb is the coefficient of kk. In this case, b=−2b = -2, so (−22)2=(−1)2=1(\frac{-2}{2})^2 = (-1)^2 = 1.
  3. Add to Both Sides: Add the number to both sides of the equation.\newlineAdd 11 to both sides of the equation to maintain equality.\newlinek2−2k+1=15+1k^2 - 2k + 1 = 15 + 1\newlinek2−2k+1=16k^2 - 2k + 1 = 16
  4. Write as Perfect Square: Write the left side of the equation as a perfect square.\newlineThe left side of the equation is now a perfect square trinomial, which can be factored into (k−1)2(k - 1)^2.\newline(k−1)2=16(k - 1)^2 = 16
  5. Take Square Root: Take the square root of both sides of the equation.\newlineTo solve for kk, take the square root of both sides.\newline(k−1)2=±16\sqrt{(k - 1)^2} = \pm\sqrt{16}\newlinek−1=±4k - 1 = \pm4
  6. Solve for k: Solve for k.\newlineAdd 11 to both sides of each equation to isolate k.\newlinek−1+1=4+1k - 1 + 1 = 4 + 1\newlinek=5k = 5\newlinek−1+1=−4+1k - 1 + 1 = -4 + 1\newlinek=−3k = -3

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