Solve g^2-2g-24+4

$${ g }^{ 2 } -2g-24+4$$

Factor

$$\left(g-\left(1-\sqrt{21}\right)\right)\left(g-\left(\sqrt{21}+1\right)\right)$$

Steps Using the Quadratic Formula

Add -24 and 4 to get -20.

$$factor(g^{2}-2g-20)$$

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.

$$g^{2}-2g-20=0$$

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.

$$g=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-20\right)}}{2}$$

Square -2.

$$g=\frac{-\left(-2\right)±\sqrt{4-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$g=\frac{-\left(-2\right)±\sqrt{4+80}}{2}$$

Add 4 to 80.

$$g=\frac{-\left(-2\right)±\sqrt{84}}{2}$$

Take the square root of 84.

$$g=\frac{-\left(-2\right)±2\sqrt{21}}{2}$$

The opposite of -2 is 2.

$$g=\frac{2±2\sqrt{21}}{2}$$

Now solve the equation g=\frac{2±2\sqrt{21}}{2} when ± is plus. Add 2 to 2\sqrt{21}.

$$g=\frac{2\sqrt{21}+2}{2}$$

Divide 2+2\sqrt{21} by 2.

$$g=\sqrt{21}+1$$

Now solve the equation g=\frac{2±2\sqrt{21}}{2} when ± is minus. Subtract 2\sqrt{21} from 2.

$$g=\frac{2-2\sqrt{21}}{2}$$

Divide 2-2\sqrt{21} by 2.

$$g=1-\sqrt{21}$$

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

$$g^{2}-2g-20=\left(g-\left(\sqrt{21}+1\right)\right)\left(g-\left(1-\sqrt{21}\right)\right)$$

Evaluate

$$g^{2}-2g-20$$

Solution Steps

Add -24 and 4 to get -20.

$$g^{2}-2g-20$$