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Solve by completing the square.\newlineq2−24q−31=0q^2 - 24q - 31 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlineq=q = _____ or q=q = _____

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Q. Solve by completing the square.\newlineq2−24q−31=0q^2 - 24q - 31 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlineq=q = _____ or q=q = _____
  1. Rewrite Equation: Rewrite the equation in the form of q2+bq=cq^2 + bq = c. Add 3131 to both sides to set the equation up for completing the square. q2−24q−31+31=0+31q^2 - 24q - 31 + 31 = 0 + 31 q2−24q=31q^2 - 24q = 31
  2. Add 3131: Find the number to add to both sides to complete the square.\newlineSince (−242)2=144(-\frac{24}{2})^2 = 144, add 144144 to both sides.\newlineq2−24q+144=31+144q^2 − 24q + 144 = 31 + 144\newlineq2−24q+144=175q^2 − 24q + 144 = 175
  3. Find Completing Square Number: Factor the left side of the equation.\newlineq2−24q+144=175q^2 - 24q + 144 = 175\newline(q−12)2=175(q - 12)^2 = 175
  4. Factor Left Side: Take the square root of both sides of the equation.\newline(q−12)2=175\sqrt{(q − 12)^2} = \sqrt{175}\newlineq−12=±175q − 12 = \pm\sqrt{175}
  5. Take Square Root: Isolate the variable qq.\newlineTo isolate qq, add 1212 to both sides of the equation.\newlineq−12+12=±175+12q - 12 + 12 = \pm\sqrt{175} + 12\newlineq=12±175q = 12 \pm \sqrt{175}
  6. Isolate Variable: Simplify the square root and round to the nearest hundredth if necessary. 175\sqrt{175} is approximately 13.2313.23 when rounded to the nearest hundredth. So, q=12±13.23q = 12 \pm 13.23
  7. Simplify Square Root: Find the two values of qq.q=12+13.23q = 12 + 13.23 implies q≈25.23q \approx 25.23.q=12−13.23q = 12 - 13.23 implies q≈−1.23q \approx -1.23.Values of qq: 25.2325.23, −1.23-1.23

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