Questions: Use the quadratic formula to solve the equation (16 x^2-40 x+74=0). Enter multiple answers as a list separated by commas. Example: (2+2 i, 2-2 i)

Solution Steps

To solve the quadratic equation \(16x^2 - 40x + 74 = 0\) using the quadratic formula, we first identify the coefficients \(a = 16\), \(b = -40\), and \(c = 74\). The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

We will calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots and then apply the formula to find the solutions.

The given quadratic equation is \(16x^2 - 40x + 74 = 0\). We identify the coefficients as follows:

• \(a = 16\)
• \(b = -40\)
• \(c = 74\)

We calculate the discriminant using the formula \(D = b^2 - 4ac\): \[ D = (-40)^2 - 4 \cdot 16 \cdot 74 = 1600 - 4736 = -3136 \] Since the discriminant is negative, the equation has two complex conjugate solutions.

Using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), we find the solutions: \[ x_1 = \frac{-(-40) + \sqrt{-3136}}{2 \cdot 16} = \frac{40 + 56i}{32} = 1.25 + 1.75i \] \[ x_2 = \frac{-(-40) - \sqrt{-3136}}{2 \cdot 16} = \frac{40 - 56i}{32} = 1.25 - 1.75i \]

Final Answer