Solve x^2-9<5x^2-12x

$${ x }^{ 2 } -9 < 5 { x }^{ 2 } -12x$$

Solve for x

$$x\neq \frac{3}{2}$$

Steps Using the Quadratic Formula

Subtract 5x^{2} from both sides.

$$x^{2}-9-5x^{2}<-12x$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-9<-12x$$

Add 12x to both sides.

$$-4x^{2}-9+12x<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -4x^{2}-9+12x positive. Since -1 is negative, the inequality direction is changed.

$$4x^{2}+9-12x>0$$

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.

$$4x^{2}+9-12x=0$$

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 4 for a, -12 for b, and 9 for c in the quadratic formula.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\times 9}}{2\times 4}$$

Do the calculations.

$$x=\frac{12±0}{8}$$

Solutions are the same.

$$x=\frac{3}{2}$$

Rewrite the inequality by using the obtained solutions.

$$4\left(x-\frac{3}{2}\right)^{2}>0$$

Inequality holds for x\neq \frac{3}{2}.

$$x\neq \frac{3}{2}$$