Solve (-4g-2-g^2)+(3g^2+9g)

$$( - 4 g - 2 - g ^ { 2 } ) + ( 3 g ^ { 2 } + 9 g )$$

Evaluate

$$2g^{2}+5g-2$$

Solution Steps

Combine -g^{2} and 3g^{2} to get 2g^{2}.

$$-4g-2+2g^{2}+9g$$

Combine -4g and 9g to get 5g.

$$5g-2+2g^{2}$$

Factor

$$2\left(g-\frac{-\sqrt{41}-5}{4}\right)\left(g-\frac{\sqrt{41}-5}{4}\right)$$

Steps Using the Quadratic Formula

Combine -g^{2} and 3g^{2} to get 2g^{2}.

$$factor(-4g-2+2g^{2}+9g)$$

Combine -4g and 9g to get 5g.

$$factor(5g-2+2g^{2})$$

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.

$$2g^{2}+5g-2=0$$

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.

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

Square 5.

$$g=\frac{-5±\sqrt{25-4\times 2\left(-2\right)}}{2\times 2}$$

Multiply -4 times 2.

$$g=\frac{-5±\sqrt{25-8\left(-2\right)}}{2\times 2}$$

Multiply -8 times -2.

$$g=\frac{-5±\sqrt{25+16}}{2\times 2}$$

Add 25 to 16.

$$g=\frac{-5±\sqrt{41}}{2\times 2}$$

Multiply 2 times 2.

$$g=\frac{-5±\sqrt{41}}{4}$$

Now solve the equation g=\frac{-5±\sqrt{41}}{4} when ± is plus. Add -5 to \sqrt{41}.

$$g=\frac{\sqrt{41}-5}{4}$$

Now solve the equation g=\frac{-5±\sqrt{41}}{4} when ± is minus. Subtract \sqrt{41} from -5.

$$g=\frac{-\sqrt{41}-5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{41}}{4} for x_{1} and \frac{-5-\sqrt{41}}{4} for x_{2}.

$$2g^{2}+5g-2=2\left(g-\frac{\sqrt{41}-5}{4}\right)\left(g-\frac{-\sqrt{41}-5}{4}\right)$$