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Question Solve the equation for all real solutions in simplest form. t^(2)+3t-19=-5

Question\newlineSolve the equation for all real solutions in simplest form.\newlinet2+3t−19=−5 t^{2}+3 t-19=-5

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Quadratic equation with complex rootsright chevron icon

Full solution

Q. Question\newlineSolve the equation for all real solutions in simplest form.\newlinet2+3t−19=−5 t^{2}+3 t-19=-5
  1. Factor Quadratic Equation: Now, we need to factor the quadratic equation or use the quadratic formula to find the roots.\newlineThe quadratic formula is (−b±b2−4ac)/(2a)(-b \pm \sqrt{b^2 - 4ac}) / (2a).\newlineHere, a=1a = 1, b=3b = 3, and c=−14c = -14.
  2. Use Quadratic Formula: Let's plug the values into the quadratic formula.\newline(−3±32−4⋅1⋅(−14))/(2⋅1)(-3 \pm \sqrt{3^2 - 4\cdot1\cdot(-14)}) / (2\cdot1)\newline(−3±9+56)/2(-3 \pm \sqrt{9 + 56}) / 2\newline(−3±65)/2(-3 \pm \sqrt{65}) / 2
  3. Plug Values: Now, simplify the square root and the fraction.\newline(−3±65)/2(-3 \pm \sqrt{65}) / 2\newlineThis gives us two solutions: (−3+65)/2(-3 + \sqrt{65}) / 2 and (−3−65)/2(-3 - \sqrt{65}) / 2
  4. Simplify Solution: Let's check if we made any mistakes.\newlineWe started with t2+3t−19=−5t^2 + 3t - 19 = -5, moved all terms to one side, used the quadratic formula correctly, and simplified.\newlineNo mistakes were made.

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