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Solve using the quadratic formula.\newline7w2−6w−9=07w^2 - 6w - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinew=w = _____ or w=w = _____

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Q. Solve using the quadratic formula.\newline7w2−6w−9=07w^2 - 6w - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinew=w = _____ or w=w = _____
  1. Identify coefficients: Identify the coefficients aa, bb, and cc in the quadratic equation 7w2−6w−9=07w^2 − 6w − 9 = 0.a=7a = 7, b=−6b = -6, c=−9c = -9
  2. Write quadratic formula: Write down the quadratic formula: w=−b±b2−4ac2aw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  3. Substitute values: Substitute the values of aa, bb, and cc into the quadratic formula.w=−(−6)±(−6)2−4⋅7⋅(−9)2⋅7w = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 7 \cdot (-9)}}{2 \cdot 7}
  4. Calculate discriminant: Simplify the equation by calculating the discriminant (the part under the square root).\newlineDiscriminant = (−6)2−4⋅7⋅(−9)=36+252=288(-6)^2 - 4\cdot7\cdot(-9) = 36 + 252 = 288
  5. Insert discriminant: Insert the discriminant back into the quadratic formula.\newlinew=6±28814w = \frac{6 \pm \sqrt{288}}{14}
  6. Simplify square root: Simplify the square root of the discriminant. 288=144×2=12×2\sqrt{288} = \sqrt{144\times 2} = 12\times\sqrt{2}
  7. Substitute square root: Substitute the simplified square root back into the equation. w=6±12214w = \frac{6 \pm 12\sqrt{2}}{14}
  8. Split into solutions: Simplify the equation by splitting it into two possible solutions. \newlinew=6+12214w = \frac{6 + 12\sqrt{2}}{14} or w=6−12214w = \frac{6 - 12\sqrt{2}}{14}
  9. Reduce fractions: Reduce the fractions to simplest form if possible.\newlinew=3+627w = \frac{3 + 6\sqrt{2}}{7} or w=3−627w = \frac{3 - 6\sqrt{2}}{7}
  10. Round solutions: If necessary, round the solutions to the nearest hundredth.\newlinew≈(3+6×1.41)/7w \approx (3 + 6 \times 1.41) / 7 or w≈(3−6×1.41)/7w \approx (3 - 6 \times 1.41) / 7\newlinew≈(3+8.46)/7w \approx (3 + 8.46) / 7 or w≈(3−8.46)/7w \approx (3 - 8.46) / 7\newlinew≈11.46/7w \approx 11.46 / 7 or w≈−5.46/7w \approx -5.46 / 7\newlinew≈1.64w \approx 1.64 or w≈−0.78w \approx -0.78

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