Question

what are the roots of the equation?
$6x^2 - 11x - 2 = 0$
answer
$\frac{11pm isqrt{73}}{12}$
$\frac{1}{6}$ and $-2$
$2$ and $-\frac{1}{6}$
$\frac{-11pmsqrt{73}}{12}$

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 6\), \(b=-11\), \(c = -2\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-(-11)\pm\sqrt{(-11)^2 - 4\times6\times(-2)}}{2\times6}\)

Step3: Calculate discriminant

\(\Delta=(-11)^2 - 4\times6\times(-2)=121 + 48 = 169\)? Wait, no, wait: Wait, \(4ac = 4\times6\times(-2)=-48\), so \(-4ac = 48\), so \(\Delta=121 + 48 = 169\)? Wait, no, original equation is \(6x^2-11x - 2 = 0\), so \(b=-11\), so \(b^2=121\), \(4ac=4\times6\times(-2)=-48\), so \(b^2 - 4ac=121-(-48)=121 + 48 = 169\)? Wait, but 169 is 13². Wait, but let's check factoring. Alternatively, maybe I miscalculated. Wait, let's try factoring. \(6x^2-11x - 2\). We need two numbers that multiply to \(6\times(-2)=-12\) and add to -11. Wait, -12 and +1: -12 +1=-11. So split middle term: \(6x^2-12x + x - 2 = 0\). Factor: \(6x(x - 2)+1(x - 2)=0\)? No, \(6x(x - 2)+1(x - 2)=(6x + 1)(x - 2)=0\)? Wait, no: \(6x(x - 2)=6x²-12x\), then +x -2: \(6x²-12x + x -2=6x²-11x -2\). Then factor: \((6x + 1)(x - 2)=0\)? Wait, no: \((6x + 1)(x - 2)=6x²-12x + x -2=6x²-11x -2\). Yes! So set to zero: \(6x + 1 = 0\) or \(x - 2 = 0\). So \(x=-\frac{1}{6}\) or \(x = 2\). Wait, so the roots are 2 and \(-\frac{1}{6}\). Wait, but let's check quadratic formula again. \(x=\frac{11\pm\sqrt{121 + 48}}{12}=\frac{11\pm\sqrt{169}}{12}=\frac{11\pm13}{12}\). So two solutions: \(\frac{11 + 13}{12}=\frac{24}{12}=2\), and \(\frac{11 - 13}{12}=\frac{-2}{12}=-\frac{1}{6}\). Ah, I see, earlier discriminant calculation: \(b^2 - 4ac=(-11)^2 - 4\times6\times(-2)=121 + 48 = 169\), which is 13². So quadratic formula: \(x=\frac{11\pm13}{12}\). So first solution: \(\frac{11 + 13}{12}=\frac{24}{12}=2\). Second: \(\frac{11 - 13}{12}=\frac{-2}{12}=-\frac{1}{6}\). So the roots are 2 and \(-\frac{1}{6}\).

Answer:

2 and \(-\frac{1}{6}\) (the option with 2 and \(-\frac{1}{6}\))