Solve (x-201)(x-203)+1=0

$$(x-201)(x-203)+1=0$$

Solve for x

$$x=202$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-201 by x-203 and combine like terms.

$$x^{2}-404x+40803+1=0$$

Add 40803 and 1 to get 40804.

$$x^{2}-404x+40804=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -404 for b, and 40804 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-404\right)±\sqrt{\left(-404\right)^{2}-4\times 40804}}{2}$$

Square -404.

$$x=\frac{-\left(-404\right)±\sqrt{163216-4\times 40804}}{2}$$

Multiply -4 times 40804.

$$x=\frac{-\left(-404\right)±\sqrt{163216-163216}}{2}$$

Add 163216 to -163216.

$$x=\frac{-\left(-404\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-404}{2}$$

The opposite of -404 is 404.

$$x=\frac{404}{2}$$

Divide 404 by 2.

$$x=202$$

Steps for Completing the Square

Use the distributive property to multiply x-201 by x-203 and combine like terms.

$$x^{2}-404x+40803+1=0$$

Add 40803 and 1 to get 40804.

$$x^{2}-404x+40804=0$$

Factor x^{2}-404x+40804. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-202\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-202\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-202=0 x-202=0$$

Add 202 to both sides of the equation.

$$x=202 x=202$$

The equation is now solved. Solutions are the same.

$$x=202$$