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Solve by completing the square.\newlinep2−8p=23p^2 - 8p = 23\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.\newlinep2−8p=23p^2 - 8p = 23\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. We have the equation p2−8p=23p^2 - 8p = 23. To complete the square, we need to move the constant term to the right side of the equation. Add 2323 to both sides to isolate the p2p^2 and pp terms. p2−8p+0=23+0p^2 - 8p + 0 = 23 + 0 p2−8p=23p^2 - 8p = 23
  2. Add Constant Term: Choose the number to add to both sides to complete the square.\newlineTo complete the square, we need to add (b/2)2(b/2)^2 to both sides, where bb is the coefficient of pp. In this case, b=−8b = -8.\newline(−8/2)2=(−4)2=16(-8/2)^2 = (-4)^2 = 16\newlineAdd 1616 to both sides of the equation.\newlinep2−8p+16=23+16p^2 − 8p + 16 = 23 + 16\newlinep2−8p+16=39p^2 − 8p + 16 = 39
  3. Choose Number to Add: Factor the left side of the equation.\newlineThe left side of the equation is now a perfect square trinomial.\newline(p−4)2=39(p - 4)^2 = 39
  4. Factor Left Side: Take the square root of both sides of the equation.\newlineTo solve for pp, take the square root of both sides.\newline(p−4)2=±39\sqrt{(p - 4)^2} = \pm\sqrt{39}\newlinep−4=±39p - 4 = \pm\sqrt{39}
  5. Take Square Root: Solve for pp.\newlineTo isolate pp, add 44 to both sides of the equation.\newlinep−4+4=±39+4p - 4 + 4 = \pm\sqrt{39} + 4\newlinep = 4±394 \pm \sqrt{39}
  6. Solve for pp: Approximate the square root of 3939 to the nearest hundredth.39\sqrt{39} is approximately 6.246.24 (rounded to the nearest hundredth). So, p≈4±6.24p \approx 4 \pm 6.24
  7. Approximate Square Root: Find the two values of pp.p≈4+6.24p \approx 4 + 6.24 implies p≈10.24p \approx 10.24.p≈4−6.24p \approx 4 - 6.24 implies p≈−2.24p \approx -2.24.Values of pp: 10.2410.24, −2.24-2.24

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