Solve (6w^2-w-5)+(4w^2-3w+2)

$$( 6 w ^ { 2 } - w - 5 ) + ( 4 w ^ { 2 } - 3 w + 2 )$$

Evaluate

$$10w^{2}-4w-3$$

Solution Steps

Combine 6w^{2} and 4w^{2} to get 10w^{2}.

$$10w^{2}-w-5-3w+2$$

Combine -w and -3w to get -4w.

$$10w^{2}-4w-5+2$$

Add -5 and 2 to get -3.

$$10w^{2}-4w-3$$

Factor

$$10\left(w-\left(-\frac{\sqrt{34}}{10}+\frac{1}{5}\right)\right)\left(w-\left(\frac{\sqrt{34}}{10}+\frac{1}{5}\right)\right)$$

Steps Using the Quadratic Formula

Combine 6w^{2} and 4w^{2} to get 10w^{2}.

$$factor(10w^{2}-w-5-3w+2)$$

Combine -w and -3w to get -4w.

$$factor(10w^{2}-4w-5+2)$$

Add -5 and 2 to get -3.

$$factor(10w^{2}-4w-3)$$

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.

$$10w^{2}-4w-3=0$$

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.

$$w=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 10\left(-3\right)}}{2\times 10}$$

Square -4.

$$w=\frac{-\left(-4\right)±\sqrt{16-4\times 10\left(-3\right)}}{2\times 10}$$

Multiply -4 times 10.

$$w=\frac{-\left(-4\right)±\sqrt{16-40\left(-3\right)}}{2\times 10}$$

Multiply -40 times -3.

$$w=\frac{-\left(-4\right)±\sqrt{16+120}}{2\times 10}$$

Add 16 to 120.

$$w=\frac{-\left(-4\right)±\sqrt{136}}{2\times 10}$$

Take the square root of 136.

$$w=\frac{-\left(-4\right)±2\sqrt{34}}{2\times 10}$$

The opposite of -4 is 4.

$$w=\frac{4±2\sqrt{34}}{2\times 10}$$

Multiply 2 times 10.

$$w=\frac{4±2\sqrt{34}}{20}$$

Now solve the equation w=\frac{4±2\sqrt{34}}{20} when ± is plus. Add 4 to 2\sqrt{34}.

$$w=\frac{2\sqrt{34}+4}{20}$$

Divide 4+2\sqrt{34} by 20.

$$w=\frac{\sqrt{34}}{10}+\frac{1}{5}$$

Now solve the equation w=\frac{4±2\sqrt{34}}{20} when ± is minus. Subtract 2\sqrt{34} from 4.

$$w=\frac{4-2\sqrt{34}}{20}$$

Divide 4-2\sqrt{34} by 20.

$$w=-\frac{\sqrt{34}}{10}+\frac{1}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{5}+\frac{\sqrt{34}}{10} for x_{1} and \frac{1}{5}-\frac{\sqrt{34}}{10} for x_{2}.

$$10w^{2}-4w-3=10\left(w-\left(\frac{\sqrt{34}}{10}+\frac{1}{5}\right)\right)\left(w-\left(-\frac{\sqrt{34}}{10}+\frac{1}{5}\right)\right)$$