Solve -x^2-8x-17<0

$$- x ^ { 2 } - 8 x - 17 < 0$$

Solve for x

$$x\in \mathrm{R}$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -x^{2}-8x-17 positive. Since -1 is negative, the inequality direction is changed.

$$x^{2}+8x+17>0$$

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.

$$x^{2}+8x+17=0$$

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, 8 for b, and 17 for c in the quadratic formula.

$$x=\frac{-8±\sqrt{8^{2}-4\times 1\times 17}}{2}$$

Do the calculations.

$$x=\frac{-8±\sqrt{-4}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression x^{2}+8x+17 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$0^{2}+8\times 0+17=17$$

The value of the expression x^{2}+8x+17 is always positive. Inequality holds for x\in \mathrm{R}.

$$x\in \mathrm{R}$$