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Solve using the quadratic formula.\newliney2−6y−9=0y^2 - 6y - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliney=y = _____ or y=y = _____

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Q. Solve using the quadratic formula.\newliney2−6y−9=0y^2 - 6y - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliney=y = _____ or y=y = _____
  1. Identify Coefficients: Identify the coefficients of the quadratic equation y2−6y−9=0y^2 - 6y - 9 = 0 by comparing it to the standard form ax2+bx+c=0ax^2 + bx + c = 0. Here, a=1a = 1, b=−6b = -6, and c=−9c = -9.
  2. Substitute Values: Substitute the values of aa, bb, and cc into the quadratic formula, which is y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\newlineSo we have y=−(−6)±(−6)2−4⋅1⋅(−9)2⋅1y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4\cdot1\cdot(-9)}}{2\cdot1}.
  3. Calculate Discriminant: Simplify the equation by calculating the discriminant (the part under the square root) first: (−6)2−4⋅1⋅(−9)=36+36=72(-6)^2 - 4\cdot 1\cdot (-9) = 36 + 36 = 72.
  4. Substitute Discriminant: Now, substitute the discriminant back into the quadratic formula: y=6±722y = \frac{6 \pm \sqrt{72}}{2}.
  5. Simplify Square Root: Simplify the square root of 7272. Since 72=36×272 = 36 \times 2 and the square root of 3636 is 66, we have 72=36×2=6×2\sqrt{72} = \sqrt{36 \times 2} = 6 \times \sqrt{2}.
  6. Substitute Simplified Square Root: Substitute the simplified square root back into the equation: y=6±6×22y = \frac{6 \pm 6 \times \sqrt{2}}{2}.
  7. Divide by 22: Divide both terms in the numerator by 22 to simplify the equation: y=3±3×22y = \frac{3 \pm 3 \times \sqrt{2}}{2}.
  8. Calculate Approximate Values: Calculate the approximate decimal values of yy by evaluating both the positive and negative cases, rounding to the nearest hundredth if necessary.\newlineFor the positive case: y≈3+3×1.41≈3+4.23≈7.23y \approx 3 + 3 \times 1.41 \approx 3 + 4.23 \approx 7.23.\newlineFor the negative case: y≈3−3×1.41≈3−4.23≈−1.23y \approx 3 - 3 \times 1.41 \approx 3 - 4.23 \approx -1.23.

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