Solve -3x^2+7x+12

$$-3 { x }^{ 2 } +7x+12$$

Factor

$$-3\left(x-\frac{7-\sqrt{193}}{6}\right)\left(x-\frac{\sqrt{193}+7}{6}\right)$$

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.

$$-3x^{2}+7x+12=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-7±\sqrt{49+12\times 12}}{2\left(-3\right)}$$

Multiply 12 times 12.

$$x=\frac{-7±\sqrt{49+144}}{2\left(-3\right)}$$

Add 49 to 144.

$$x=\frac{-7±\sqrt{193}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±\sqrt{193}}{-6}$$

Now solve the equation x=\frac{-7±\sqrt{193}}{-6} when ± is plus. Add -7 to \sqrt{193}.

$$x=\frac{\sqrt{193}-7}{-6}$$

Divide -7+\sqrt{193} by -6.

$$x=\frac{7-\sqrt{193}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{193}}{-6} when ± is minus. Subtract \sqrt{193} from -7.

$$x=\frac{-\sqrt{193}-7}{-6}$$

Divide -7-\sqrt{193} by -6.

$$x=\frac{\sqrt{193}+7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{193}}{6} for x_{1} and \frac{7+\sqrt{193}}{6} for x_{2}.

$$-3x^{2}+7x+12=-3\left(x-\frac{7-\sqrt{193}}{6}\right)\left(x-\frac{\sqrt{193}+7}{6}\right)$$

Evaluate

$$12+7x-3x^{2}$$