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Solve using the quadratic formula.\newline4k2−7k+2=04k^2 - 7k + 2 = 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.\newline4k2−7k+2=04k^2 - 7k + 2 = 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 Coefficients: Identify the coefficients of the quadratic equation 4k2−7k+2=04k^2 - 7k + 2 = 0. The standard form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0. By comparing, we find that a=4a = 4, b=−7b = -7, and c=2c = 2.
  2. Substitute Values: Substitute the values of aa, bb, and cc into the quadratic formula, k=−b±b2−4ac2ak = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Here, a=4a = 4, b=−7b = -7, and c=2c = 2, so we substitute these into the formula to get k=−(−7)±(−7)2−4⋅4⋅22⋅4k = \frac{-(-7) \pm \sqrt{(-7)^2 - 4\cdot4\cdot2}}{2\cdot4}.
  3. Simplify Expression: Simplify the expression inside the square root and the constants outside the square root.\newlineThe expression inside the square root becomes 49−32\sqrt{49 - 32} and the constants outside become 2×42 \times 4.\newlineSo, k=7±178k = \frac{7 \pm \sqrt{17}}{8}.
  4. Calculate Solutions: Calculate the two possible solutions for kk using the simplified quadratic formula.\newlineThe first solution is k=7+178k = \frac{7 + \sqrt{17}}{8} and the second solution is k=7−178k = \frac{7 - \sqrt{17}}{8}.
  5. Round Values: Round the values of kk to the nearest hundredth, if necessary.\newlineThe first solution is approximately k≈(7+4.12)/8≈11.12/8≈1.39k \approx (7 + 4.12) / 8 \approx 11.12 / 8 \approx 1.39.\newlineThe second solution is approximately k≈(7−4.12)/8≈2.88/8≈0.36k \approx (7 - 4.12) / 8 \approx 2.88 / 8 \approx 0.36.

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