Solve c^2-c+3/2>0

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

Solve for c

$$c\in \mathrm{R}$$

Steps Using the Quadratic Formula

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.

$$c^{2}-c+\frac{3}{2}=0$$

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

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

Do the calculations.

$$c=\frac{1±\sqrt{-5}}{2}$$

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

$$0^{2}-0+\frac{3}{2}=\frac{3}{2}$$

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

$$c\in \mathrm{R}$$