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Solve using the quadratic formula.\newline7t2+t−5=07t^2 + t - 5 = 0\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 using the quadratic formula.\newline7t2+t−5=07t^2 + t - 5 = 0\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. Identify Coefficients: Identify the coefficients of the quadratic equation.\newlineThe quadratic equation is in the form at2+bt+c=0at^2 + bt + c = 0. For the equation 7t2+t−5=07t^2 + t - 5 = 0, the coefficients are:\newlinea=7a = 7, b=1b = 1, and c=−5c = -5.
  2. Write Quadratic Formula: Write down the quadratic formula.\newlineThe quadratic formula is t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  3. Substitute into Formula: Substitute the coefficients into the quadratic formula.\newlinet=−(1)±(1)2−4(7)(−5)2(7)t = \frac{-(1) \pm \sqrt{(1)^2 - 4(7)(-5)}}{2(7)}\newlinet=−1±1+14014t = \frac{-1 \pm \sqrt{1 + 140}}{14}\newlinet=−1±14114t = \frac{-1 \pm \sqrt{141}}{14}
  4. Simplify and Calculate Discriminant: Simplify under the square root and calculate the discriminant. 141\sqrt{141} is an irrational number, so we will leave it under the square root.
  5. Calculate Possible Values: Calculate the two possible values for tt.t=−1+14114t = \frac{{-1 + \sqrt{141}}}{{14}} or t=−1−14114t = \frac{{-1 - \sqrt{141}}}{{14}}
  6. Simplify Fractions: Simplify the fractions if possible.\newlineThe fractions cannot be simplified further, so we will leave them as is.
  7. Round to Nearest Hundredth: Round the decimal values to the nearest hundredth if necessary.\newlineSince the question asks for integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth, we will provide the solutions in the form of improper fractions.\newlinet=−1+14114t = \frac{-1 + \sqrt{141}}{14} or t=−1−14114t = \frac{-1 - \sqrt{141}}{14}

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