Solve k^2+5k+8>0

$$k ^ { 2 } + 5 k + 8 > 0$$

Solve for k

$$k\in \mathrm{R}$$

Steps Using the Quadratic Formula

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.

$$k^{2}+5k+8=0$$

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 1 for a, 5 for b, and 8 for c in the quadratic formula.

$$k=\frac{-5±\sqrt{5^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$k=\frac{-5±\sqrt{-7}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression k^{2}+5k+8 has the same sign for any k. To determine the sign, calculate the value of the expression for k=0.

$$0^{2}+5\times 0+8=8$$

The value of the expression k^{2}+5k+8 is always positive. Inequality holds for k\in \mathrm{R}.

$$k\in \mathrm{R}$$