Question

1. $4x^2 - 1 = -8x$

Explanation:

Step1: Rearrange the equation to standard quadratic form.

We start with the equation \(4x^{2}-1 = -8x\). To get it into the standard form \(ax^{2}+bx + c=0\), we add \(8x\) to both sides:
\(4x^{2}+8x - 1=0\)

Step2: Identify coefficients \(a\), \(b\), and \(c\).

For the quadratic equation \(4x^{2}+8x - 1 = 0\), we have \(a = 4\), \(b = 8\), and \(c=- 1\).

Step3: Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

First, calculate the discriminant \(\Delta=b^{2}-4ac\). Substitute \(a = 4\), \(b = 8\), and \(c=-1\) into the discriminant formula:
\(\Delta=(8)^{2}-4\times4\times(-1)=64 + 16=80\)

Then, substitute \(a = 4\), \(b = 8\), and \(\Delta = 80\) into the quadratic formula:
\(x=\frac{-8\pm\sqrt{80}}{2\times4}=\frac{-8\pm4\sqrt{5}}{8}=\frac{-2\pm\sqrt{5}}{2}\)

Answer:

\(x=\frac{-2 + \sqrt{5}}{2}\) or \(x=\frac{-2-\sqrt{5}}{2}\)