Solve 2^3-32^2*b+5*2*b^2=

$$2 ^ { 3 } - 32 ^ { 2 } \cdot b + 5 \cdot 2 \cdot b ^ { 2 } =$$

Evaluate

$$10b^{2}-1024b+8$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

$$8-32^{2}b+5\times 2b^{2}$$

Calculate 32 to the power of 2 and get 1024.

$$8-1024b+5\times 2b^{2}$$

Multiply 5 and 2 to get 10.

$$8-1024b+10b^{2}$$

Factor

$$10\left(b-\frac{256-2\sqrt{16379}}{5}\right)\left(b-\frac{2\sqrt{16379}+256}{5}\right)$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 3 and get 8.

$$factor(8-32^{2}b+5\times 2b^{2})$$

Calculate 32 to the power of 2 and get 1024.

$$factor(8-1024b+5\times 2b^{2})$$

Multiply 5 and 2 to get 10.

$$factor(8-1024b+10b^{2})$$

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.

$$10b^{2}-1024b+8=0$$

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.

$$b=\frac{-\left(-1024\right)±\sqrt{\left(-1024\right)^{2}-4\times 10\times 8}}{2\times 10}$$

Square -1024.

$$b=\frac{-\left(-1024\right)±\sqrt{1048576-4\times 10\times 8}}{2\times 10}$$

Multiply -4 times 10.

$$b=\frac{-\left(-1024\right)±\sqrt{1048576-40\times 8}}{2\times 10}$$

Multiply -40 times 8.

$$b=\frac{-\left(-1024\right)±\sqrt{1048576-320}}{2\times 10}$$

Add 1048576 to -320.

$$b=\frac{-\left(-1024\right)±\sqrt{1048256}}{2\times 10}$$

Take the square root of 1048256.

$$b=\frac{-\left(-1024\right)±8\sqrt{16379}}{2\times 10}$$

The opposite of -1024 is 1024.

$$b=\frac{1024±8\sqrt{16379}}{2\times 10}$$

Multiply 2 times 10.

$$b=\frac{1024±8\sqrt{16379}}{20}$$

Now solve the equation b=\frac{1024±8\sqrt{16379}}{20} when ± is plus. Add 1024 to 8\sqrt{16379}.

$$b=\frac{8\sqrt{16379}+1024}{20}$$

Divide 1024+8\sqrt{16379} by 20.

$$b=\frac{2\sqrt{16379}+256}{5}$$

Now solve the equation b=\frac{1024±8\sqrt{16379}}{20} when ± is minus. Subtract 8\sqrt{16379} from 1024.

$$b=\frac{1024-8\sqrt{16379}}{20}$$

Divide 1024-8\sqrt{16379} by 20.

$$b=\frac{256-2\sqrt{16379}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{256+2\sqrt{16379}}{5} for x_{1} and \frac{256-2\sqrt{16379}}{5} for x_{2}.

$$10b^{2}-1024b+8=10\left(b-\frac{2\sqrt{16379}+256}{5}\right)\left(b-\frac{256-2\sqrt{16379}}{5}\right)$$