Solve x^2+2x+3>3/2x

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

Solve for x

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

Steps Using the Quadratic Formula

Subtract \frac{3}{2}x from both sides.

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

Combine 2x and -\frac{3}{2}x to get \frac{1}{2}x.

$$x^{2}+\frac{1}{2}x+3>0$$

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.

$$x^{2}+\frac{1}{2}x+3=0$$

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{1}{2} for b, and 3 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{-\frac{1}{2}±\sqrt{-\frac{47}{4}}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression x^{2}+\frac{1}{2}x+3 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$0^{2}+\frac{1}{2}\times 0+3=3$$

The value of the expression x^{2}+\frac{1}{2}x+3 is always positive. Inequality holds for x\in \mathrm{R}.

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