Solve 3x^2-5x+9>0

$$3 { x }^{ 2 } -5x+9 > 0$$

Solve for x

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

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.

$$3x^{2}-5x+9=0$$

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 3 for a, -5 for b, and 9 for c in the quadratic formula.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\times 9}}{2\times 3}$$

Do the calculations.

$$x=\frac{5±\sqrt{-83}}{6}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 3x^{2}-5x+9 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$3\times 0^{2}-5\times 0+9=9$$

The value of the expression 3x^{2}-5x+9 is always positive. Inequality holds for x\in \mathrm{R}.

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