Solve 69x-4<2x^2

$$69 x - 4 < 2 x ^ { 2 }$$

Solve for x

$$x\in \left(-\infty,\frac{69-\sqrt{4729}}{4}\right)\cup \left(\frac{\sqrt{4729}+69}{4},\infty\right)$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$69x-4-2x^{2}<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 69x-4-2x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-69x+4+2x^{2}>0$$

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.

$$-69x+4+2x^{2}=0$$

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 2 for a, -69 for b, and 4 for c in the quadratic formula.

$$x=\frac{-\left(-69\right)±\sqrt{\left(-69\right)^{2}-4\times 2\times 4}}{2\times 2}$$

Do the calculations.

$$x=\frac{69±\sqrt{4729}}{4}$$

Solve the equation x=\frac{69±\sqrt{4729}}{4} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{4729}+69}{4} x=\frac{69-\sqrt{4729}}{4}$$

Rewrite the inequality by using the obtained solutions.

$$2\left(x-\frac{\sqrt{4729}+69}{4}\right)\left(x-\frac{69-\sqrt{4729}}{4}\right)>0$$

For the product to be positive, x-\frac{\sqrt{4729}+69}{4} and x-\frac{69-\sqrt{4729}}{4} have to be both negative or both positive. Consider the case when x-\frac{\sqrt{4729}+69}{4} and x-\frac{69-\sqrt{4729}}{4} are both negative.

$$x-\frac{\sqrt{4729}+69}{4}<0 x-\frac{69-\sqrt{4729}}{4}<0$$

The solution satisfying both inequalities is x<\frac{69-\sqrt{4729}}{4}.

$$x<\frac{69-\sqrt{4729}}{4}$$

Consider the case when x-\frac{\sqrt{4729}+69}{4} and x-\frac{69-\sqrt{4729}}{4} are both positive.

$$x-\frac{69-\sqrt{4729}}{4}>0 x-\frac{\sqrt{4729}+69}{4}>0$$

The solution satisfying both inequalities is x>\frac{\sqrt{4729}+69}{4}.

$$x>\frac{\sqrt{4729}+69}{4}$$

The final solution is the union of the obtained solutions.

$$x<\frac{69-\sqrt{4729}}{4}\text{; }x>\frac{\sqrt{4729}+69}{4}$$