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Solve using the quadratic formula.\newlinet2+2t+1=0t^2 + 2t + 1 = 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.\newlinet2+2t+1=0t^2 + 2t + 1 = 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. Quadratic Formula Coefficients: The quadratic formula is given by t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients from the quadratic equation at2+bt+c=0at^2 + bt + c = 0. In this case, a=1a = 1, b=2b = 2, and c=1c = 1.
  2. Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. For our equation, the discriminant is (2)2−4(1)(1)=4−4=0(2)^2 - 4(1)(1) = 4 - 4 = 0.
  3. One Real Solution: Since the discriminant is 00, there is only one real solution to the equation. This is because the square root of 00 is 00, and −b±0-b \pm 0 is simply −b-b.
  4. Apply Quadratic Formula: Now, apply the quadratic formula with the discriminant being 00: t=−2±02⋅1=−2±02t = \frac{-2 \pm \sqrt{0}}{2 \cdot 1} = \frac{-2 \pm 0}{2}.
  5. Simplify Expression: Simplify the expression: t=(−2)/2=−1t = (-2) / 2 = -1.
  6. Unique Solution: Since the discriminant was 00, there is only one unique solution to the equation, which is t=−1t = -1.

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