Solve {l}{3n^2-6n-}{-140>0}

$$\left. \begin{array} { l } { 3 n ^ { 2 } - 6 n - } \\ { - 140 > 0 } \end{array} \right.$$

Solve for n

$$n\in \mathrm{R}$$

Steps Using the Quadratic Formula

The opposite of -140 is 140.

$$3n^{2}-6n+140>0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$3n^{2}-6n+140=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 3 for a, -6 for b, and 140 for c in the quadratic formula.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\times 140}}{2\times 3}$$

Do the calculations.

$$n=\frac{6±\sqrt{-1644}}{6}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 3n^{2}-6n+140 has the same sign for any n. To determine the sign, calculate the value of the expression for n=0.

$$3\times 0^{2}-6\times 0+140=140$$

The value of the expression 3n^{2}-6n+140 is always positive. Inequality holds for n\in \mathrm{R}.

$$n\in \mathrm{R}$$