Solve 36-2^2-22b-b^2

$$36 - 2 ^ { 2 } - 22 b - b ^ { 2 }$$

Evaluate

$$32-22b-b^{2}$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$36-4-22b-b^{2}$$

Subtract 4 from 36 to get 32.

$$32-22b-b^{2}$$

Factor

$$-\left(b-\left(-3\sqrt{17}-11\right)\right)\left(b-\left(3\sqrt{17}-11\right)\right)$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 2 and get 4.

$$factor(36-4-22b-b^{2})$$

Subtract 4 from 36 to get 32.

$$factor(32-22b-b^{2})$$

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.

$$-b^{2}-22b+32=0$$

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.

$$b=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Square -22.

$$b=\frac{-\left(-22\right)±\sqrt{484-4\left(-1\right)\times 32}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-\left(-22\right)±\sqrt{484+4\times 32}}{2\left(-1\right)}$$

Multiply 4 times 32.

$$b=\frac{-\left(-22\right)±\sqrt{484+128}}{2\left(-1\right)}$$

Add 484 to 128.

$$b=\frac{-\left(-22\right)±\sqrt{612}}{2\left(-1\right)}$$

Take the square root of 612.

$$b=\frac{-\left(-22\right)±6\sqrt{17}}{2\left(-1\right)}$$

The opposite of -22 is 22.

$$b=\frac{22±6\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{22±6\sqrt{17}}{-2}$$

Now solve the equation b=\frac{22±6\sqrt{17}}{-2} when ± is plus. Add 22 to 6\sqrt{17}.

$$b=\frac{6\sqrt{17}+22}{-2}$$

Divide 22+6\sqrt{17} by -2.

$$b=-3\sqrt{17}-11$$

Now solve the equation b=\frac{22±6\sqrt{17}}{-2} when ± is minus. Subtract 6\sqrt{17} from 22.

$$b=\frac{22-6\sqrt{17}}{-2}$$

Divide 22-6\sqrt{17} by -2.

$$b=3\sqrt{17}-11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -11-3\sqrt{17} for x_{1} and -11+3\sqrt{17} for x_{2}.

$$-b^{2}-22b+32=-\left(b-\left(-3\sqrt{17}-11\right)\right)\left(b-\left(3\sqrt{17}-11\right)\right)$$