nswers to two decimal places. 2. \( -5 x^{2}-12 x+16 \)

We begin with the quadratic equation \[ -5x^2 - 12x + 16 = 0. \] **Step 1. Multiply by \(-1\) to simplify the equation:** \[ 5x^2 + 12x - 16 = 0. \] **Step 2. Apply the quadratic formula.** For a quadratic equation of the form \[ ax^2 + bx + c = 0, \] the solutions are given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] Here, \(a = 5\), \(b = 12\), and \(c = -16\). **Step 3. Compute the discriminant:** \[ \Delta = b^2 - 4ac = 12^2 - 4(5)(-16) = 144 + 320 = 464. \] **Step 4. Write the solutions:** \[ x = \frac{-12 \pm \sqrt{464}}{2 \cdot 5} = \frac{-12 \pm \sqrt{464}}{10}. \] Notice that \(\sqrt{464}\) can be simplified: \[ \sqrt{464} = \sqrt{16 \cdot 29} = 4\sqrt{29}. \] Thus, the solutions become \[ x = \frac{-12 \pm 4\sqrt{29}}{10} = \frac{-6 \pm 2\sqrt{29}}{5}. \] **Step 5. Approximate the solutions to two decimal places.** First, compute \(\sqrt{29}\). Using a calculator: \[ \sqrt{29} \approx 5.39. \] Then, \[ 2\sqrt{29} \approx 2 \times 5.39 = 10.78. \] Now find the two solutions: 1. For the positive case: \[ x = \frac{-6 + 10.78}{5} = \frac{4.78}{5} \approx 0.96. \] 2. For the negative case: \[ x = \frac{-6 - 10.78}{5} = \frac{-16.78}{5} \approx -3.36. \] Thus, the solutions rounded to two decimal places are: \[ x \approx 0.96 \quad \text{or} \quad x \approx -3.36. \]