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Solve by completing the square.\newlinew2−18w=31w^2 - 18w = 31\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinew=w = _____ or w=w = _____

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Q. Solve by completing the square.\newlinew2−18w=31w^2 - 18w = 31\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinew=w = _____ or w=w = _____
  1. Write Equation Form: Write the equation in the form of w2+bw=cw^2 + bw = c. The given equation is already in this form: w2−18w=31w^2 - 18w = 31.
  2. Move Constant Term: Move the constant term to the right side of the equation.\newlineAdd 3131 to both sides to isolate the ww terms on one side.\newlinew2−18w+(−31+31)=31+(−31)w^2 − 18w + (−31 + 31) = 31 + (−31)\newlinew2−18w=31w^2 − 18w = 31
  3. Find Completing Number: Find the number to complete the square.\newlineTake half of the coefficient of ww, square it, and add it to both sides of the equation.\newline(−18/2)2=81(-18/2)^2 = 81\newlinew2−18w+81=31+81w^2 − 18w + 81 = 31 + 81\newlinew2−18w+81=112w^2 − 18w + 81 = 112
  4. Factor Left Side: Factor the left side of the equation.\newlineThe left side is now a perfect square trinomial.\newline(w−9)2=112(w - 9)^2 = 112
  5. Take Square Root: Take the square root of both sides of the equation.\newline(w−9)2=±112\sqrt{(w - 9)^2} = \pm\sqrt{112}\newlinew−9=±112w - 9 = \pm\sqrt{112}
  6. Solve for w: Solve for w.\newlineAdd 99 to both sides of the equation to isolate ww.\newlinew=9±112w = 9 \pm \sqrt{112}\newlinew=9±(16×7)w = 9 \pm \sqrt{(16 \times 7)}\newlinew=9±47w = 9 \pm 4\sqrt{7}\newlineSince 7\sqrt{7} is approximately 2.652.65, we can also express the solutions as decimals.\newlinew≈9+4(2.65)w \approx 9 + 4(2.65) or w≈9−4(2.65)w \approx 9 - 4(2.65)\newlinew≈9+10.6w \approx 9 + 10.6 or ww00\newlineww11 or ww22

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