Solve x^2-180x+3870

$${ x }^{ 2 } -180x+3870$$

Factor

$$\left(x-\left(90-3\sqrt{470}\right)\right)\left(x-\left(3\sqrt{470}+90\right)\right)$$

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.

$$x^{2}-180x+3870=0$$

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.

$$x=\frac{-\left(-180\right)±\sqrt{\left(-180\right)^{2}-4\times 3870}}{2}$$

Square -180.

$$x=\frac{-\left(-180\right)±\sqrt{32400-4\times 3870}}{2}$$

Multiply -4 times 3870.

$$x=\frac{-\left(-180\right)±\sqrt{32400-15480}}{2}$$

Add 32400 to -15480.

$$x=\frac{-\left(-180\right)±\sqrt{16920}}{2}$$

Take the square root of 16920.

$$x=\frac{-\left(-180\right)±6\sqrt{470}}{2}$$

The opposite of -180 is 180.

$$x=\frac{180±6\sqrt{470}}{2}$$

Now solve the equation x=\frac{180±6\sqrt{470}}{2} when ± is plus. Add 180 to 6\sqrt{470}.

$$x=\frac{6\sqrt{470}+180}{2}$$

Divide 180+6\sqrt{470} by 2.

$$x=3\sqrt{470}+90$$

Now solve the equation x=\frac{180±6\sqrt{470}}{2} when ± is minus. Subtract 6\sqrt{470} from 180.

$$x=\frac{180-6\sqrt{470}}{2}$$

Divide 180-6\sqrt{470} by 2.

$$x=90-3\sqrt{470}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 90+3\sqrt{470} for x_{1} and 90-3\sqrt{470} for x_{2}.

$$x^{2}-180x+3870=\left(x-\left(3\sqrt{470}+90\right)\right)\left(x-\left(90-3\sqrt{470}\right)\right)$$

Evaluate

$$x^{2}-180x+3870$$