Solve 5x^2+9<6x^2+7x

$$5 { x }^{ 2 } +9 < 6 { x }^{ 2 } +7x$$

Solve for x

$$x\in \left(-\infty,\frac{-\sqrt{85}-7}{2}\right)\cup \left(\frac{\sqrt{85}-7}{2},\infty\right)$$

Steps Using the Quadratic Formula

Subtract 6x^{2} from both sides.

$$5x^{2}+9-6x^{2}<7x$$

Combine 5x^{2} and -6x^{2} to get -x^{2}.

$$-x^{2}+9<7x$$

Subtract 7x from both sides.

$$-x^{2}+9-7x<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -x^{2}+9-7x positive. Since -1 is negative, the inequality direction is changed.

$$x^{2}-9+7x>0$$

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.

$$x^{2}-9+7x=0$$

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, 7 for b, and -9 for c in the quadratic formula.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\left(-9\right)}}{2}$$

Do the calculations.

$$x=\frac{-7±\sqrt{85}}{2}$$

Solve the equation x=\frac{-7±\sqrt{85}}{2} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{85}-7}{2} x=\frac{-\sqrt{85}-7}{2}$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-\frac{\sqrt{85}-7}{2}\right)\left(x-\frac{-\sqrt{85}-7}{2}\right)>0$$

For the product to be positive, x-\frac{\sqrt{85}-7}{2} and x-\frac{-\sqrt{85}-7}{2} have to be both negative or both positive. Consider the case when x-\frac{\sqrt{85}-7}{2} and x-\frac{-\sqrt{85}-7}{2} are both negative.

$$x-\frac{\sqrt{85}-7}{2}<0 x-\frac{-\sqrt{85}-7}{2}<0$$

The solution satisfying both inequalities is x<\frac{-\sqrt{85}-7}{2}.

$$x<\frac{-\sqrt{85}-7}{2}$$

Consider the case when x-\frac{\sqrt{85}-7}{2} and x-\frac{-\sqrt{85}-7}{2} are both positive.

$$x-\frac{-\sqrt{85}-7}{2}>0 x-\frac{\sqrt{85}-7}{2}>0$$

The solution satisfying both inequalities is x>\frac{\sqrt{85}-7}{2}.

$$x>\frac{\sqrt{85}-7}{2}$$

The final solution is the union of the obtained solutions.

$$x<\frac{-\sqrt{85}-7}{2}\text{; }x>\frac{\sqrt{85}-7}{2}$$