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Solve by completing the square.\newlinex2−22x−31=0x^2 - 22x - 31 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____

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Q. Solve by completing the square.\newlinex2−22x−31=0x^2 - 22x - 31 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____
  1. Rewrite equation: Rewrite the equation in the form of x2+bx=cx^2 + bx = c.\newlineAdd 3131 to both sides to set the equation up for completing the square.\newlinex2−22x−31+31=0+31x^2 - 22x - 31 + 31 = 0 + 31\newlinex2−22x=31x^2 - 22x = 31
  2. Add 3131: Choose the number to add to both sides to complete the square.\newlineSince (−222)2=121(-\frac{22}{2})^2 = 121, add 121121 to both sides.\newlinex2−22x+121=31+121x^2 − 22x + 121 = 31 + 121\newlinex2−22x+121=152x^2 − 22x + 121 = 152
  3. Choose number to add: Factor the left side of the equation.\newlinex2−22x+121=152x^2 - 22x + 121 = 152\newline(x−11)2=152(x - 11)^2 = 152
  4. Factor left side: Take the square root of both sides of the equation.\newline(x−11)2=152\sqrt{(x − 11)^2} = \sqrt{152}\newlinex−11=±152x − 11 = \pm\sqrt{152}
  5. Take square root: Solve for xx by isolating the variable.\newlineTo isolate xx, add 1111 to both sides of the equation.\newlinex−11+11=±152+11x - 11 + 11 = \pm\sqrt{152} + 11\newlinex=11±152x = 11 \pm \sqrt{152}
  6. Solve for x: Calculate the approximate values of x.\newlinex≈11±12.33x \approx 11 \pm 12.33 (since 152\sqrt{152} is approximately 12.3312.33)\newlinex≈11+12.33x \approx 11 + 12.33 implies x≈23.33x \approx 23.33.\newlinex≈11−12.33x \approx 11 - 12.33 implies x≈−1.33x \approx -1.33.\newlineValues of x: 23.3323.33, −1.33-1.33

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