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Solve by completing the square.\newlinet2−28t=−13t^2 - 28t = -13\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinet=t = _____ or t=t = _____

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Q. Solve by completing the square.\newlinet2−28t=−13t^2 - 28t = -13\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinet=t = _____ or t=t = _____
  1. Write Equation Form: Write the equation in the form of t2+bt=ct^2 + bt = c. The given equation is t2−28t=−13t^2 - 28t = -13. We want to complete the square on the left side.
  2. Add 1313 and Isolate: Add 1313 to both sides to isolate the t2t^2 and tt terms on one side.\newlinet2−28t+13=0t^2 - 28t + 13 = 0
  3. Complete the Square: To complete the square, we need to add (b2)2(\frac{b}{2})^2 to both sides, where bb is the coefficient of tt. In this case, b=−28b = -28, so (b2)2=(−282)2=(−14)2=196(\frac{b}{2})^2 = (\frac{-28}{2})^2 = (-14)^2 = 196.t2−28t+196=13+196t^2 − 28t + 196 = 13 + 196
  4. Simplify Right Side: Simplify the right side of the equation. t2−28t+196=209t^2 - 28t + 196 = 209
  5. Factor as Perfect Square: Factor the left side as a perfect square.\newline(t−14)2=209(t - 14)^2 = 209
  6. Take Square Root: Take the square root of both sides to solve for tt.(t−14)2=±209\sqrt{(t - 14)^2} = \pm\sqrt{209}t−14=±209t - 14 = \pm\sqrt{209}
  7. Isolate and Solve for t: Isolate tt by adding 1414 to both sides.\newlinet=14±209t = 14 \pm \sqrt{209}
  8. Calculate Square Root: Calculate the square root of 209209 and round to the nearest hundredth.\newline209≈14.45\sqrt{209} \approx 14.45\newlinet≈14±14.45t \approx 14 \pm 14.45
  9. Find Values of t: Find the two values of t.\newlinet≈14+14.45t \approx 14 + 14.45 implies t≈28.45t \approx 28.45.\newlinet≈14−14.45t \approx 14 - 14.45 implies t≈−0.45t \approx -0.45.\newlineValues of t: 28.4528.45, −0.45-0.45

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