Solve 0<x^2+3/4x+2

$$0 < x ^ { 2 } + \frac { 3 } { 4 } x + 2$$

Solve for x

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

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side. This changes the sign direction.

$$x^{2}+\frac{3}{4}x+2>0$$

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.

$$x^{2}+\frac{3}{4}x+2=0$$

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, \frac{3}{4} for b, and 2 for c in the quadratic formula.

$$x=\frac{-\frac{3}{4}±\sqrt{\left(\frac{3}{4}\right)^{2}-4\times 1\times 2}}{2}$$

Do the calculations.

$$x=\frac{-\frac{3}{4}±\sqrt{-\frac{119}{16}}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression x^{2}+\frac{3}{4}x+2 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$0^{2}+\frac{3}{4}\times 0+2=2$$

The value of the expression x^{2}+\frac{3}{4}x+2 is always positive. Inequality holds for x\in \mathrm{R}.

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