Solve 12+9*x-5x^2

$$12+9 \times x-5 { x }^{ 2 }$$

Factor

$$-5\left(x-\frac{9-\sqrt{321}}{10}\right)\left(x-\frac{\sqrt{321}+9}{10}\right)$$

Steps Using the Quadratic Formula

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.

$$-5x^{2}+9x+12=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-5\right)\times 12}}{2\left(-5\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-5\right)\times 12}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-9±\sqrt{81+20\times 12}}{2\left(-5\right)}$$

Multiply 20 times 12.

$$x=\frac{-9±\sqrt{81+240}}{2\left(-5\right)}$$

Add 81 to 240.

$$x=\frac{-9±\sqrt{321}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-9±\sqrt{321}}{-10}$$

Now solve the equation x=\frac{-9±\sqrt{321}}{-10} when ± is plus. Add -9 to \sqrt{321}.

$$x=\frac{\sqrt{321}-9}{-10}$$

Divide -9+\sqrt{321} by -10.

$$x=\frac{9-\sqrt{321}}{10}$$

Now solve the equation x=\frac{-9±\sqrt{321}}{-10} when ± is minus. Subtract \sqrt{321} from -9.

$$x=\frac{-\sqrt{321}-9}{-10}$$

Divide -9-\sqrt{321} by -10.

$$x=\frac{\sqrt{321}+9}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9-\sqrt{321}}{10} for x_{1} and \frac{9+\sqrt{321}}{10} for x_{2}.

$$-5x^{2}+9x+12=-5\left(x-\frac{9-\sqrt{321}}{10}\right)\left(x-\frac{\sqrt{321}+9}{10}\right)$$

Evaluate

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