Question

ratic formula to solve. express your answer in simple
$4z^2 - 4z - 15 = 0$

Explanation:

Step1: Recall Quadratic Formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For the equation \(4z^{2}-4z - 15=0\), we have \(a = 4\), \(b=-4\), \(c=-15\).

Step2: Calculate Discriminant

First, calculate the discriminant \(D=b^{2}-4ac\). Substitute the values: \(D=(-4)^{2}-4\times4\times(-15)=16 + 240=256\).

Step3: Apply Quadratic Formula

Now, substitute \(a\), \(b\), and \(D\) into the quadratic formula: \(z=\frac{-(-4)\pm\sqrt{256}}{2\times4}=\frac{4\pm16}{8}\).

Step4: Find Two Solutions

For the plus sign: \(z=\frac{4 + 16}{8}=\frac{20}{8}=\frac{5}{2}\). For the minus sign: \(z=\frac{4-16}{8}=\frac{-12}{8}=-\frac{3}{2}\).

Answer:

The solutions are \(z=\frac{5}{2}\) and \(z =-\frac{3}{2}\)