Question

2x^{2}-x - 6 = 0

Explanation:

Step1: Identify coefficients

For the quadratic equation \(2x^{2}-x - 6=0\), \(a = 2\), \(b=-1\), \(c = - 6\).

Step2: Use quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 2\), \(b=-1\), \(c=-6\) into it:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-1)^{2}-4\times2\times(-6)=1 + 48=49\).
Then \(x=\frac{-(-1)\pm\sqrt{49}}{2\times2}=\frac{1\pm7}{4}\).

Step3: Find two solutions

When we take the plus - sign: \(x=\frac{1 + 7}{4}=\frac{8}{4}=2\).
When we take the minus - sign: \(x=\frac{1-7}{4}=\frac{-6}{4}=-\frac{3}{2}\).

Answer:

\(x = 2\) or \(x=-\frac{3}{2}\)