Solve (5w^2-w-4)+(-6w+2)

$$( 5 w ^ { 2 } - w - 4 ) + ( - 6 w + 2 )$$

Evaluate

$$5w^{2}-7w-2$$

Solution Steps

Combine -w and -6w to get -7w.

$$5w^{2}-7w-4+2$$

Add -4 and 2 to get -2.

$$5w^{2}-7w-2$$

Factor

$$5\left(w-\frac{7-\sqrt{89}}{10}\right)\left(w-\frac{\sqrt{89}+7}{10}\right)$$

Steps Using the Quadratic Formula

Combine -w and -6w to get -7w.

$$factor(5w^{2}-7w-4+2)$$

Add -4 and 2 to get -2.

$$factor(5w^{2}-7w-2)$$

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.

$$5w^{2}-7w-2=0$$

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.

$$w=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square -7.

$$w=\frac{-\left(-7\right)±\sqrt{49-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply -4 times 5.

$$w=\frac{-\left(-7\right)±\sqrt{49-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$w=\frac{-\left(-7\right)±\sqrt{49+40}}{2\times 5}$$

Add 49 to 40.

$$w=\frac{-\left(-7\right)±\sqrt{89}}{2\times 5}$$

The opposite of -7 is 7.

$$w=\frac{7±\sqrt{89}}{2\times 5}$$

Multiply 2 times 5.

$$w=\frac{7±\sqrt{89}}{10}$$

Now solve the equation w=\frac{7±\sqrt{89}}{10} when ± is plus. Add 7 to \sqrt{89}.

$$w=\frac{\sqrt{89}+7}{10}$$

Now solve the equation w=\frac{7±\sqrt{89}}{10} when ± is minus. Subtract \sqrt{89} from 7.

$$w=\frac{7-\sqrt{89}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{89}}{10} for x_{1} and \frac{7-\sqrt{89}}{10} for x_{2}.

$$5w^{2}-7w-2=5\left(w-\frac{\sqrt{89}+7}{10}\right)\left(w-\frac{7-\sqrt{89}}{10}\right)$$