Solution - Solving quadratic inequalities using the quadratic formula

The coefficients of our inequality, $$x^2+0x+322\ge 0$$, are:

$$a$$ = 1

$$b$$ = 0

$$c$$ = 322

To find the roots of a quadratic equation, plug its coefficients ($$a$$, $$b$$ and $$c$$ ) into the quadratic formula:

$$x=\left(-b\pm sqrt\left(b^2-4ac\right)\right)/\left(2a\right)$$

$$x=\left(-0\pm sqrt\left(0^2-4*1*322\right)\right)/\left(2*1\right)$$

$$x=\left(-0\pm sqrt\left(0-4*1*322\right)\right)/\left(2*1\right)$$

$$x=\left(-0\pm sqrt\left(0-4*322\right)\right)/\left(2*1\right)$$

$$x=\left(-0\pm sqrt\left(0-1288\right)\right)/\left(2*1\right)$$

$$x=\left(-0\pm sqrt\left(-1288\right)\right)/\left(2*1\right)$$

$$x=\left(-0\pm sqrt\left(-1288\right)\right)/\left(2\right)$$

to get the result:

$$x=\left(-0\pm sqrt\left(-1288\right)\right)/2$$

Simplify $$-1288$$ by finding its prime factors:

The prime factorization of $$-1288$$ is $$2i·\sqrt{322}$$

$$x=\left(-0\pm 2i*sqrt\left(322\right)\right)/2$$

The ± means two roots are possible.

Separate the equations:
$$x_1=\left(-0+2i*sqrt\left(322\right)\right)/2$$ and $$x_2=\left(-0-2i*sqrt\left(322\right)\right)/2$$

$$x_1=\frac{2i·\sqrt{322}}{2}$$

Simplify the fraction:

$$x_1=i·\sqrt{322}$$

$$x_2=\frac{-2i·\sqrt{322}}{2}$$

Simplify the fraction:

$$x_2=-i·\sqrt{322}$$

Discriminant part of the quadratic formula:

$$b^2-4ac<0$$ There are no real roots.
$$b^2-4ac=0$$ There is one real root.
$$b^2-4ac>0$$ There are two real roots.

Inequality function has no real roots, the parabola does not intersect with the x-axis. The quadratic formula requires taking the square root, and square root of negative number is not defined over the real line.

Interval is $$\left(-\infty ,\infty \right)$$