Solve x^2-19x+84<0

$${ x }^{ 2 } -19x+84 < 0$$

Solve for x

$$x\in \left(7,12\right)$$

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.

$$x^{2}-19x+84=0$$

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, -19 for b, and 84 for c in the quadratic formula.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 1\times 84}}{2}$$

Do the calculations.

$$x=\frac{19±5}{2}$$

Solve the equation x=\frac{19±5}{2} when ± is plus and when ± is minus.

$$x=12 x=7$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-12\right)\left(x-7\right)<0$$

For the product to be negative, x-12 and x-7 have to be of the opposite signs. Consider the case when x-12 is positive and x-7 is negative.

$$x-12>0 x-7<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-7 is positive and x-12 is negative.

$$x-7>0 x-12<0$$

The solution satisfying both inequalities is x\in \left(7,12\right).

$$x\in \left(7,12\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(7,12\right)$$