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Solve using the quadratic formula.\newline8w2−3w−9=08w^2 - 3w - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newline`w=`____ or `w=`____

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Q. Solve using the quadratic formula.\newline8w2−3w−9=08w^2 - 3w - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newline`w=`____ or `w=`____
  1. Identify Coefficients: Identify the coefficients of the quadratic equation 8w2−3w−9=08w^2 - 3w - 9 = 0 to use in the quadratic formula. The standard form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0. Here, a=8a = 8, b=−3b = -3, and c=−9c = -9.
  2. Write Quadratic Formula: Write down the quadratic formula, which is w=−b±b2−4ac2aw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. We will substitute the values of aa, bb, and cc into this formula to find the solutions for ww.
  3. Substitute Values: Substitute the values of aa, bb, and cc into the quadratic formula: w=−(−3)±(−3)2−4⋅8⋅(−9)2⋅8w = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 8 \cdot (-9)}}{2 \cdot 8}.
  4. Calculate Discriminant: Simplify the equation by calculating the discriminant (the part under the square root): (−3)2−4⋅8⋅(−9)=9+288=297\sqrt{(-3)^2 - 4 \cdot 8 \cdot (-9)} = \sqrt{9 + 288} = \sqrt{297}.
  5. Calculate Solutions: Now, calculate the two possible solutions for ww using the simplified discriminant: w=3±29716w = \frac{3 \pm \sqrt{297}}{16}.
  6. Simplify Solutions: Simplify the solutions further. Since 297\sqrt{297} cannot be simplified to an integer or a simple fraction, we will use a calculator to find the decimal approximation: 297≈17.23\sqrt{297} \approx 17.23.
  7. Calculate Decimal Approximation: Calculate the two solutions for ww by adding and subtracting the approximate value of 297\sqrt{297}: w≈(3+17.23)/16w \approx (3 + 17.23) / 16 or w≈(3−17.23)/16w \approx (3 - 17.23) / 16.
  8. Perform Division: Perform the division to find the approximate decimal values: w≈20.2316w \approx \frac{20.23}{16} or w≈−14.2316w \approx \frac{-14.23}{16}, which gives w≈1.27w \approx 1.27 or w≈−0.89w \approx -0.89 when rounded to the nearest hundredth.

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