Solve 3w^2-8w-20

$$3 { w }^{ 2 } -8w-20$$

Factor

$$3\left(w-\frac{4-2\sqrt{19}}{3}\right)\left(w-\frac{2\sqrt{19}+4}{3}\right)$$

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.

$$3w^{2}-8w-20=0$$

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.

$$w=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\left(-20\right)}}{2\times 3}$$

Square -8.

$$w=\frac{-\left(-8\right)±\sqrt{64-4\times 3\left(-20\right)}}{2\times 3}$$

Multiply -4 times 3.

$$w=\frac{-\left(-8\right)±\sqrt{64-12\left(-20\right)}}{2\times 3}$$

Multiply -12 times -20.

$$w=\frac{-\left(-8\right)±\sqrt{64+240}}{2\times 3}$$

Add 64 to 240.

$$w=\frac{-\left(-8\right)±\sqrt{304}}{2\times 3}$$

Take the square root of 304.

$$w=\frac{-\left(-8\right)±4\sqrt{19}}{2\times 3}$$

The opposite of -8 is 8.

$$w=\frac{8±4\sqrt{19}}{2\times 3}$$

Multiply 2 times 3.

$$w=\frac{8±4\sqrt{19}}{6}$$

Now solve the equation w=\frac{8±4\sqrt{19}}{6} when ± is plus. Add 8 to 4\sqrt{19}.

$$w=\frac{4\sqrt{19}+8}{6}$$

Divide 8+4\sqrt{19} by 6.

$$w=\frac{2\sqrt{19}+4}{3}$$

Now solve the equation w=\frac{8±4\sqrt{19}}{6} when ± is minus. Subtract 4\sqrt{19} from 8.

$$w=\frac{8-4\sqrt{19}}{6}$$

Divide 8-4\sqrt{19} by 6.

$$w=\frac{4-2\sqrt{19}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4+2\sqrt{19}}{3} for x_{1} and \frac{4-2\sqrt{19}}{3} for x_{2}.

$$3w^{2}-8w-20=3\left(w-\frac{2\sqrt{19}+4}{3}\right)\left(w-\frac{4-2\sqrt{19}}{3}\right)$$

Evaluate

$$3w^{2}-8w-20$$