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Solve using the quadratic formula.\newline6k2+6k+1=06k^2 + 6k + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____

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Q. Solve using the quadratic formula.\newline6k2+6k+1=06k^2 + 6k + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____
  1. Identify values: Identify the values of aa, bb, and cc in the quadratic equation 6k2+6k+1=06k^2 + 6k + 1 = 0. By comparing the equation with the standard form ax2+bx+c=0ax^2 + bx + c = 0, we find: a=6a = 6 b=6b = 6 c=1c = 1
  2. Substitute into formula: Substitute the values of aa, bb, and cc into the quadratic formula to find kk. The quadratic formula is k=−b±b2−4ac2ak = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. So we have: k=−(6)±(6)2−4⋅6⋅12⋅6k = \frac{-(6) \pm \sqrt{(6)^2 - 4\cdot6\cdot1}}{2\cdot6}
  3. Simplify discriminant: Simplify the expression under the square root (the discriminant). (6)2−4⋅6⋅1=36−24=12\sqrt{(6)^2 - 4\cdot 6\cdot 1} = \sqrt{36 - 24} = \sqrt{12}
  4. Continue simplifying formula: Continue simplifying the quadratic formula with the values we have.\newlinek=−6±1212k = \frac{-6 \pm \sqrt{12}}{12}\newlineSince 12\sqrt{12} can be simplified to 232\sqrt{3}, we get:\newlinek=−6±2312k = \frac{-6 \pm 2\sqrt{3}}{12}
  5. Divide by common factor: Simplify the expression by dividing all terms by the common factor of 22.k=−3±36k = \frac{-3 \pm \sqrt{3}}{6}
  6. Identify possible values: Identify the two possible values for kk.k=−3+36k = \frac{{-3 + \sqrt{3}}}{{6}} or k=−3−36k = \frac{{-3 - \sqrt{3}}}{{6}}
  7. Round to nearest hundredth: If necessary, round the values of kk to the nearest hundredth.k≈(−3+1.736)k \approx (\frac{-3 + 1.73}{6}) or k≈(−3−1.736)k \approx (\frac{-3 - 1.73}{6})k≈−1.276k \approx \frac{-1.27}{6} or k≈−4.736k \approx \frac{-4.73}{6}k≈−0.21k \approx -0.21 or k≈−0.79k \approx -0.79

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