Solve -4x^2+3x+12

$$-4 { x }^{ 2 } +3x+12$$

Factor

$$-4\left(x-\frac{3-\sqrt{201}}{8}\right)\left(x-\frac{\sqrt{201}+3}{8}\right)$$

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.

$$-4x^{2}+3x+12=0$$

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.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-4\right)\times 12}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-3±\sqrt{9+16\times 12}}{2\left(-4\right)}$$

Multiply 16 times 12.

$$x=\frac{-3±\sqrt{9+192}}{2\left(-4\right)}$$

Add 9 to 192.

$$x=\frac{-3±\sqrt{201}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-3±\sqrt{201}}{-8}$$

Now solve the equation x=\frac{-3±\sqrt{201}}{-8} when ± is plus. Add -3 to \sqrt{201}.

$$x=\frac{\sqrt{201}-3}{-8}$$

Divide -3+\sqrt{201} by -8.

$$x=\frac{3-\sqrt{201}}{8}$$

Now solve the equation x=\frac{-3±\sqrt{201}}{-8} when ± is minus. Subtract \sqrt{201} from -3.

$$x=\frac{-\sqrt{201}-3}{-8}$$

Divide -3-\sqrt{201} by -8.

$$x=\frac{\sqrt{201}+3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{201}}{8} for x_{1} and \frac{3+\sqrt{201}}{8} for x_{2}.

$$-4x^{2}+3x+12=-4\left(x-\frac{3-\sqrt{201}}{8}\right)\left(x-\frac{\sqrt{201}+3}{8}\right)$$

Evaluate

$$12+3x-4x^{2}$$