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Solve by completing the square.\newlinef2−22f=−37f^2 - 22f = -37\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinef=f = _____ or f=f = _____

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Q. Solve by completing the square.\newlinef2−22f=−37f^2 - 22f = -37\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinef=f = _____ or f=f = _____
  1. Write Equation in Form: Write the equation in the form of f2+bf=cf^2 + bf = c. The given equation is f2−22f=−37f^2 - 22f = -37. We want to complete the square on the left side.
  2. Add 3737 to Both Sides: Add 3737 to both sides to isolate the quadratic term and the linear term on one side.\newlinef2−22f+37=0f^2 - 22f + 37 = 0
  3. Find Completing Square Number: Find the number that completes the square. This is found by taking half of the coefficient of ff, squaring it, and adding it to both sides.\newlineThe coefficient of ff is −22-22, so half of −22-22 is −11-11. Squaring −11-11 gives us 121121.\newlinef2−22f+121=37+121f^2 − 22f + 121 = 37 + 121
  4. Rewrite as Perfect Square Trinomial: Rewrite the equation as a perfect square trinomial on the left side.\newlinef2−22f+121=158f^2 - 22f + 121 = 158\newline(f−11)2=158(f - 11)^2 = 158
  5. Take Square Root to Solve: Take the square root of both sides to solve for ff.(f−11)2=±158\sqrt{(f - 11)^2} = \pm\sqrt{158}f−11=±158f - 11 = \pm\sqrt{158}
  6. Solve for f: Solve for f by adding 1111 to both sides.\newlinef=11±158f = 11 \pm \sqrt{158}
  7. Calculate Square Root of 158158: Calculate the square root of 158158 and round to the nearest hundredth.158≈12.57\sqrt{158} \approx 12.57f≈11±12.57f \approx 11 \pm 12.57
  8. Find Two Values of f: Find the two values of f.\newlinef≈11+12.57f \approx 11 + 12.57 implies f≈23.57f \approx 23.57.\newlinef≈11−12.57f \approx 11 - 12.57 implies f≈−1.57f \approx -1.57.\newlineValues of f: 23.5723.57, −1.57-1.57

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