Questions: Solve the following equation. 4 z^2 + 9 = 16 z z =

Solution Steps

To solve the quadratic equation \(4z^2 + 9 = 16z\), we first rearrange it into the standard form \(az^2 + bz + c = 0\). Then, we use the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(z\).

The given equation is \(4z^2 + 9 = 16z\). We rearrange it into the standard quadratic form: \[ 4z^2 - 16z + 9 = 0 \]

Identify the coefficients from the quadratic equation \(az^2 + bz + c = 0\):

• \(a = 4\)
• \(b = -16\)
• \(c = 9\)

The discriminant \(\Delta\) is calculated as: \[ \Delta = b^2 - 4ac = (-16)^2 - 4 \times 4 \times 9 = 256 - 144 = 112 \]

Using the quadratic formula \(z = \frac{-b \pm \sqrt{\Delta}}{2a}\), we find the solutions: \[ z_1 = \frac{-(-16) + \sqrt{112}}{2 \times 4} = \frac{16 + \sqrt{112}}{8} \] \[ z_2 = \frac{-(-16) - \sqrt{112}}{2 \times 4} = \frac{16 - \sqrt{112}}{8} \]

Simplify the solutions: \[ z_1 = \frac{16 + 10.5830}{8} = 3.3229 \] \[ z_2 = \frac{16 - 10.5830}{8} = 0.6771 \]

Final Answer