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Solve using the quadratic formula.\newline9j2+2j−6=09j^2 + 2j - 6 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinej=j = _____ or j=j = _____

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Q. Solve using the quadratic formula.\newline9j2+2j−6=09j^2 + 2j - 6 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinej=j = _____ or j=j = _____
  1. Identify values of aa, bb, cc: Identify the values of aa, bb, and cc in the quadratic equation 9j2+2j−6=09j^2 + 2j - 6 = 0. Compare 9j2+2j−6=09j^2 + 2j - 6 = 0 with the standard form ax2+bx+c=0ax^2 + bx + c = 0. a=9a = 9 bb00 bb11
  2. Substitute values into formula: Substitute the values of aa, bb, and cc into the quadratic formula j=−b±b2−4ac2aj = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Substitute a=9a = 9, b=2b = 2, and c=−6c = -6 into the quadratic formula. j=−(2)±(2)2−4⋅9⋅(−6)2⋅9j = \frac{-(2) \pm \sqrt{(2)^2 - 4\cdot9\cdot(-6)}}{2\cdot9}
  3. Simplify discriminant calculation: Simplify the expression under the square root, known as the discriminant.\newlineCalculate (2)2−4⋅9⋅(−6)\sqrt{(2)^2 - 4\cdot9\cdot(-6)}.\newline(2)2−4⋅9⋅(−6)\sqrt{(2)^2 - 4\cdot9\cdot(-6)}\newline= 4+216\sqrt{4 + 216}\newline= 220\sqrt{220}
  4. Simplify quadratic formula: Simplify the quadratic formula with the calculated discriminant.\newlinej=−2±2202⋅9j = \frac{-2 \pm \sqrt{220}}{2\cdot9}\newlinej=−2±22018j = \frac{-2 \pm \sqrt{220}}{18}
  5. Calculate possible solutions: Calculate the two possible solutions for jj.j=−2+22018j = \frac{{-2 + \sqrt{220}}}{{18}} or j=−2−22018j = \frac{{-2 - \sqrt{220}}}{{18}}
  6. Simplify square root of 220220: Simplify the square root of 220220 to its simplest radical form if possible. 220\sqrt{220} can be simplified to 4×55\sqrt{4\times55} which is 2×552\times\sqrt{55}. So, j=−2+2×5518j = \frac{-2 + 2\times\sqrt{55}}{18} or j=−2−2×5518j = \frac{-2 - 2\times\sqrt{55}}{18}
  7. Simplify fractions: Simplify the fractions by dividing the numerator and denominator by the common factor if possible.\newlineIn this case, there is no common factor for all terms, so we leave the fractions as they are.\newlinej=−2+25518j = \frac{-2 + 2\sqrt{55}}{18} or j=−2−25518j = \frac{-2 - 2\sqrt{55}}{18}
  8. Round values to nearest hundredth: Round the values of jj to the nearest hundredth, if required.j \approx (\-2 + 2\times7.42) / 18 or j \approx (\-2 - 2\times7.42) / 18j≈(14.84−2)/18j \approx (14.84 - 2) / 18 or j \approx (\-14.84 - 2) / 18j≈12.84/18j \approx 12.84 / 18 or j \approx \-16.84 / 18j≈0.71j \approx 0.71 or j \approx \-0.93

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