Solve -16x^2+17x+12

$$-16 { x }^{ 2 } +17x+12$$

Factor

$$-16\left(x-\frac{17-\sqrt{1057}}{32}\right)\left(x-\frac{\sqrt{1057}+17}{32}\right)$$

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.

$$-16x^{2}+17x+12=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-16\right)\times 12}}{2\left(-16\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-16\right)\times 12}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-17±\sqrt{289+64\times 12}}{2\left(-16\right)}$$

Multiply 64 times 12.

$$x=\frac{-17±\sqrt{289+768}}{2\left(-16\right)}$$

Add 289 to 768.

$$x=\frac{-17±\sqrt{1057}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-17±\sqrt{1057}}{-32}$$

Now solve the equation x=\frac{-17±\sqrt{1057}}{-32} when ± is plus. Add -17 to \sqrt{1057}.

$$x=\frac{\sqrt{1057}-17}{-32}$$

Divide -17+\sqrt{1057} by -32.

$$x=\frac{17-\sqrt{1057}}{32}$$

Now solve the equation x=\frac{-17±\sqrt{1057}}{-32} when ± is minus. Subtract \sqrt{1057} from -17.

$$x=\frac{-\sqrt{1057}-17}{-32}$$

Divide -17-\sqrt{1057} by -32.

$$x=\frac{\sqrt{1057}+17}{32}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{17-\sqrt{1057}}{32} for x_{1} and \frac{17+\sqrt{1057}}{32} for x_{2}.

$$-16x^{2}+17x+12=-16\left(x-\frac{17-\sqrt{1057}}{32}\right)\left(x-\frac{\sqrt{1057}+17}{32}\right)$$

Evaluate

$$12+17x-16x^{2}$$