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Solve by completing the square.\newlinep2−22p+45=0p^2 - 22p + 45 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinep=p = _____ or p=p = _____

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Q. Solve by completing the square.\newlinep2−22p+45=0p^2 - 22p + 45 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinep=p = _____ or p=p = _____
  1. Rewrite equation: Rewrite the equation in the form of p2+bp=cp^2 + bp = c.\newlineSubtract 4545 from both sides to set the equation up for completing the square.\newlinep2−22p+45−45=0−45p^2 − 22p + 45 - 45 = 0 - 45\newlinep2−22p=−45p^2 − 22p = -45
  2. Subtract and set up: Add the square of half the coefficient of pp to both sides to complete the square.\newlineSince (−22/2)2=121(-22/2)^2 = 121, add 121121 to both sides.\newlinep2−22p+121=−45+121p^2 − 22p + 121 = -45 + 121\newlinep2−22p+121=76p^2 − 22p + 121 = 76
  3. Add to complete square: Factor the left side of the equation.\newlinep2−22p+121p^2 - 22p + 121 is a perfect square trinomial.\newline(p−11)2=76(p - 11)^2 = 76
  4. Factor left side: Take the square root of both sides of the equation.\newline(p−11)2=±76\sqrt{(p - 11)^2} = \pm\sqrt{76}\newlinep−11=±76p - 11 = \pm\sqrt{76}
  5. Take square root: Solve for pp by adding 1111 to both sides of the equation.\newlinep−11+11=±76+11p - 11 + 11 = \pm\sqrt{76} + 11\newlinep=11±76p = 11 \pm \sqrt{76}
  6. Solve for pp: Calculate the approximate decimal values of pp.p≈11±8.72p \approx 11 \pm 8.72 (since 76\sqrt{76} is approximately 8.728.72)p≈11+8.72p \approx 11 + 8.72 or p≈11−8.72p \approx 11 - 8.72p≈19.72p \approx 19.72 or p≈2.28p \approx 2.28

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