Solve 2/3x^2+x/2+5/6>0

$$\frac { 2 } { 3 } x ^ { 2 } + \frac { x } { 2 } + \frac { 5 } { 6 } > 0$$

Solve for x

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

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2,6. Since 6 is positive, the inequality direction remains the same.

$$4x^{2}+3x+5>0$$

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.

$$4x^{2}+3x+5=0$$

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 4 for a, 3 for b, and 5 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 4\times 5}}{2\times 4}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-71}}{8}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 4x^{2}+3x+5 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$4\times 0^{2}+3\times 0+5=5$$

The value of the expression 4x^{2}+3x+5 is always positive. Inequality holds for x\in \mathrm{R}.

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