Solve 13x^2-32x+1

$$13 { x }^{ 2 } -32x+1$$

Factor

$$13\left(x-\frac{16-9\sqrt{3}}{13}\right)\left(x-\frac{9\sqrt{3}+16}{13}\right)$$

Steps Using the Quadratic Formula

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.

$$13x^{2}-32x+1=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 13}}{2\times 13}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 13}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-32\right)±\sqrt{1024-52}}{2\times 13}$$

Add 1024 to -52.

$$x=\frac{-\left(-32\right)±\sqrt{972}}{2\times 13}$$

Take the square root of 972.

$$x=\frac{-\left(-32\right)±18\sqrt{3}}{2\times 13}$$

The opposite of -32 is 32.

$$x=\frac{32±18\sqrt{3}}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{32±18\sqrt{3}}{26}$$

Now solve the equation x=\frac{32±18\sqrt{3}}{26} when ± is plus. Add 32 to 18\sqrt{3}.

$$x=\frac{18\sqrt{3}+32}{26}$$

Divide 32+18\sqrt{3} by 26.

$$x=\frac{9\sqrt{3}+16}{13}$$

Now solve the equation x=\frac{32±18\sqrt{3}}{26} when ± is minus. Subtract 18\sqrt{3} from 32.

$$x=\frac{32-18\sqrt{3}}{26}$$

Divide 32-18\sqrt{3} by 26.

$$x=\frac{16-9\sqrt{3}}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{16+9\sqrt{3}}{13} for x_{1} and \frac{16-9\sqrt{3}}{13} for x_{2}.

$$13x^{2}-32x+1=13\left(x-\frac{9\sqrt{3}+16}{13}\right)\left(x-\frac{16-9\sqrt{3}}{13}\right)$$

Evaluate

$$13x^{2}-32x+1$$