Solve (x-2020)(x-2021)=6

$$(x-2020)(x-2021)=6$$

Solve for x

$$x=2018$$

$$x=2023$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2020 by x-2021 and combine like terms.

$$x^{2}-4041x+4082420=6$$

Subtract 6 from both sides.

$$x^{2}-4041x+4082420-6=0$$

Subtract 6 from 4082420 to get 4082414.

$$x^{2}-4041x+4082414=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4041 for b, and 4082414 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4041\right)±\sqrt{\left(-4041\right)^{2}-4\times 4082414}}{2}$$

Square -4041.

$$x=\frac{-\left(-4041\right)±\sqrt{16329681-4\times 4082414}}{2}$$

Multiply -4 times 4082414.

$$x=\frac{-\left(-4041\right)±\sqrt{16329681-16329656}}{2}$$

Add 16329681 to -16329656.

$$x=\frac{-\left(-4041\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-4041\right)±5}{2}$$

The opposite of -4041 is 4041.

$$x=\frac{4041±5}{2}$$

Now solve the equation x=\frac{4041±5}{2} when ± is plus. Add 4041 to 5.

$$x=\frac{4046}{2}$$

Divide 4046 by 2.

$$x=2023$$

Now solve the equation x=\frac{4041±5}{2} when ± is minus. Subtract 5 from 4041.

$$x=\frac{4036}{2}$$

Divide 4036 by 2.

$$x=2018$$

The equation is now solved.

$$x=2023 x=2018$$

Steps for Completing the Square

Use the distributive property to multiply x-2020 by x-2021 and combine like terms.

$$x^{2}-4041x+4082420=6$$

Subtract 4082420 from both sides.

$$x^{2}-4041x=6-4082420$$

Subtract 4082420 from 6 to get -4082414.

$$x^{2}-4041x=-4082414$$

Divide -4041, the coefficient of the x term, by 2 to get -\frac{4041}{2}. Then add the square of -\frac{4041}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4041x+\left(-\frac{4041}{2}\right)^{2}=-4082414+\left(-\frac{4041}{2}\right)^{2}$$

Square -\frac{4041}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-4041x+\frac{16329681}{4}=-4082414+\frac{16329681}{4}$$

Add -4082414 to \frac{16329681}{4}.

$$x^{2}-4041x+\frac{16329681}{4}=\frac{25}{4}$$

Factor x^{2}-4041x+\frac{16329681}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4041}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4041}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{4041}{2}=\frac{5}{2} x-\frac{4041}{2}=-\frac{5}{2}$$

Add \frac{4041}{2} to both sides of the equation.

$$x=2023 x=2018$$