Solve -21x^2+16x+2

$$-21 { x }^{ 2 } +16x+2$$

Factor

$$-21\left(x-\frac{8-\sqrt{106}}{21}\right)\left(x-\frac{\sqrt{106}+8}{21}\right)$$

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.

$$-21x^{2}+16x+2=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-21\right)\times 2}}{2\left(-21\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-21\right)\times 2}}{2\left(-21\right)}$$

Multiply -4 times -21.

$$x=\frac{-16±\sqrt{256+84\times 2}}{2\left(-21\right)}$$

Multiply 84 times 2.

$$x=\frac{-16±\sqrt{256+168}}{2\left(-21\right)}$$

Add 256 to 168.

$$x=\frac{-16±\sqrt{424}}{2\left(-21\right)}$$

Take the square root of 424.

$$x=\frac{-16±2\sqrt{106}}{2\left(-21\right)}$$

Multiply 2 times -21.

$$x=\frac{-16±2\sqrt{106}}{-42}$$

Now solve the equation x=\frac{-16±2\sqrt{106}}{-42} when ± is plus. Add -16 to 2\sqrt{106}.

$$x=\frac{2\sqrt{106}-16}{-42}$$

Divide -16+2\sqrt{106} by -42.

$$x=\frac{8-\sqrt{106}}{21}$$

Now solve the equation x=\frac{-16±2\sqrt{106}}{-42} when ± is minus. Subtract 2\sqrt{106} from -16.

$$x=\frac{-2\sqrt{106}-16}{-42}$$

Divide -16-2\sqrt{106} by -42.

$$x=\frac{\sqrt{106}+8}{21}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{8-\sqrt{106}}{21} for x_{1} and \frac{8+\sqrt{106}}{21} for x_{2}.

$$-21x^{2}+16x+2=-21\left(x-\frac{8-\sqrt{106}}{21}\right)\left(x-\frac{\sqrt{106}+8}{21}\right)$$

Evaluate

$$2+16x-21x^{2}$$