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Solve using the quadratic formula.\newline3j2+8j−8=03j^2 + 8j - 8 = 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.\newline3j2+8j−8=03j^2 + 8j - 8 = 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 3j2+8j−8=03j^2 + 8j - 8 = 0. The quadratic equation is in the form aj2+bj+c=0aj^2 + bj + c = 0. Comparing this with our equation, we get: a=3a = 3 b=8b = 8 bb00
  2. Substitute into quadratic 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=3a = 3, b=8b = 8, and c=−8c = -8 into the quadratic formula. j=−(8)±(8)2−4⋅3⋅(−8)2⋅3j = \frac{-(8) \pm \sqrt{(8)^2 - 4\cdot3\cdot(-8)}}{2\cdot3}
  3. Simplify discriminant: Simplify the expression under the square root, known as the discriminant.\newlineCalculate (8)2−4⋅3⋅(−8)\sqrt{(8)^2 - 4\cdot 3\cdot (-8)}.\newline(8)2−4⋅3⋅(−8)\sqrt{(8)^2 - 4\cdot 3\cdot (-8)}\newline= 64+96\sqrt{64 + 96}\newline= 160\sqrt{160}
  4. Simplify quadratic formula: Simplify the quadratic formula with the calculated discriminant.\newlinej=−8±1602⋅3j = \frac{-8 \pm \sqrt{160}}{2 \cdot 3}\newlinej=−8±1606j = \frac{-8 \pm \sqrt{160}}{6}
  5. Calculate possible solutions: Calculate the two possible solutions for jj.\newlineFirst solution:\newlinej=−8+1606j = \frac{{-8 + \sqrt{160}}}{{6}}\newlineSecond solution:\newlinej=−8−1606j = \frac{{-8 - \sqrt{160}}}{{6}}
  6. Simplify square root: Simplify the square root of 160160 to its simplest radical form if possible.160\sqrt{160} can be simplified to 4104\sqrt{10} because 160=16×10160 = 16\times 10 and 16=4\sqrt{16} = 4. So, 160=410\sqrt{160} = 4\sqrt{10}.
  7. Substitute simplified root: Substitute the simplified square root back into the solutions.\newlineFirst solution:\newlinej=−8+4106j = \frac{-8 + 4\sqrt{10}}{6}\newlineSecond solution:\newlinej=−8−4106j = \frac{-8 - 4\sqrt{10}}{6}
  8. Simplify fractions: Simplify the fractions if possible.\newlineFirst solution:\newlinej=−8+4106j = \frac{-8 + 4\sqrt{10}}{6} can be simplified by dividing numerator and denominator by 22.\newlinej=−4+2103j = \frac{-4 + 2\sqrt{10}}{3}\newlineSecond solution:\newlinej=−8−4106j = \frac{-8 - 4\sqrt{10}}{6} can be simplified by dividing numerator and denominator by 22.\newlinej=−4−2103j = \frac{-4 - 2\sqrt{10}}{3}
  9. Round to nearest hundredth: If required, round the solutions to the nearest hundredth.\newlineFirst solution:\newlinej≈(−4+2103)≈(−4+2×3.163)≈(−4+6.323)≈2.323≈0.77j \approx (\frac{-4 + 2\sqrt{10}}{3}) \approx (\frac{-4 + 2\times3.16}{3}) \approx (\frac{-4 + 6.32}{3}) \approx \frac{2.32}{3} \approx 0.77\newlineSecond solution:\newlinej≈(−4−2103)≈(−4−2×3.163)≈(−4−6.323)≈−10.323≈−3.44j \approx (\frac{-4 - 2\sqrt{10}}{3}) \approx (\frac{-4 - 2\times3.16}{3}) \approx (\frac{-4 - 6.32}{3}) \approx \frac{-10.32}{3} \approx -3.44

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