Questions: Solve the equation using factoring or the quadratic formula, whichever is appropriate. Answer in lowest terms. -3z + 1 = -4z^2 Hint: Answer in the form (a ± b i sqrt(c))/d and reduce all terms. Enter 1 for b or d if they are 1.

Solution Steps

To solve the quadratic equation \(-3z + 1 = -4z^2\), we first need to rearrange it into the standard form \(az^2 + bz + c = 0\). Then, we can use the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

1. Rearrange the equation to standard form.
2. Identify coefficients \(a\), \(b\), and \(c\).
3. Apply the quadratic formula to find the solutions.
4. Simplify the solutions to the required form.

Rearrange the given equation \(-3z + 1 = -4z^2\) into the standard quadratic form \(az^2 + bz + c = 0\).

\[ -4z^2 + 3z + 1 = 0 \]

Identify the coefficients \(a\), \(b\), and \(c\) from the standard form equation:

\[ a = 4, \quad b = -3, \quad c = 1 \]

Use the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

\[ z = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]

Simplify the expression under the square root and the entire formula:

\[ z = \frac{3 \pm \sqrt{9 - 16}}{8} = \frac{3 \pm \sqrt{-7}}{8} = \frac{3 \pm i\sqrt{7}}{8} \]

Final Answer