Solve (2019x)^2+2018*2020x=1

$$( 2019 x ) ^ { 2 } + 2018 \times 2020 x = 1$$

Solve for x

$$x=-1$$

$$x=\frac{1}{4076361}\approx 0.000000245$$

Steps Using Factoring By Grouping

Expand \left(2019x\right)^{2}.

$$2019^{2}x^{2}+2018\times 2020x=1$$

Calculate 2019 to the power of 2 and get 4076361.

$$4076361x^{2}+2018\times 2020x=1$$

Multiply 2018 and 2020 to get 4076360.

$$4076361x^{2}+4076360x=1$$

Subtract 1 from both sides.

$$4076361x^{2}+4076360x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4076361x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=4076360 ab=4076361\left(-1\right)=-4076361$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4076361.

$$-1,4076361 -3,1358787 -9,452929 -673,6057 -2019,2019$$

Calculate the sum for each pair.

$$-1+4076361=4076360 -3+1358787=1358784 -9+452929=452920 -673+6057=5384 -2019+2019=0$$

The solution is the pair that gives sum 4076360.

$$a=-1 b=4076361$$

Rewrite 4076361x^{2}+4076360x-1 as \left(4076361x^{2}-x\right)+\left(4076361x-1\right).

$$\left(4076361x^{2}-x\right)+\left(4076361x-1\right)$$

Factor out x in 4076361x^{2}-x.

$$x\left(4076361x-1\right)+4076361x-1$$

Factor out common term 4076361x-1 by using distributive property.

$$\left(4076361x-1\right)\left(x+1\right)$$

To find equation solutions, solve 4076361x-1=0 and x+1=0.

$$x=\frac{1}{4076361} x=-1$$

Steps Using the Quadratic Formula

Expand \left(2019x\right)^{2}.

$$2019^{2}x^{2}+2018\times 2020x=1$$

Calculate 2019 to the power of 2 and get 4076361.

$$4076361x^{2}+2018\times 2020x=1$$

Multiply 2018 and 2020 to get 4076360.

$$4076361x^{2}+4076360x=1$$

Subtract 1 from both sides.

$$4076361x^{2}+4076360x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4076361 for a, 4076360 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4076360±\sqrt{4076360^{2}-4\times 4076361\left(-1\right)}}{2\times 4076361}$$

Square 4076360.

$$x=\frac{-4076360±\sqrt{16616710849600-4\times 4076361\left(-1\right)}}{2\times 4076361}$$

Multiply -4 times 4076361.

$$x=\frac{-4076360±\sqrt{16616710849600-16305444\left(-1\right)}}{2\times 4076361}$$

Multiply -16305444 times -1.

$$x=\frac{-4076360±\sqrt{16616710849600+16305444}}{2\times 4076361}$$

Add 16616710849600 to 16305444.

$$x=\frac{-4076360±\sqrt{16616727155044}}{2\times 4076361}$$

Take the square root of 16616727155044.

$$x=\frac{-4076360±4076362}{2\times 4076361}$$

Multiply 2 times 4076361.

$$x=\frac{-4076360±4076362}{8152722}$$

Now solve the equation x=\frac{-4076360±4076362}{8152722} when ± is plus. Add -4076360 to 4076362.

$$x=\frac{2}{8152722}$$

Reduce the fraction \frac{2}{8152722} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{4076361}$$

Now solve the equation x=\frac{-4076360±4076362}{8152722} when ± is minus. Subtract 4076362 from -4076360.

$$x=-\frac{8152722}{8152722}$$

Divide -8152722 by 8152722.

$$x=-1$$

The equation is now solved.

$$x=\frac{1}{4076361} x=-1$$

Steps for Completing the Square

Expand \left(2019x\right)^{2}.

$$2019^{2}x^{2}+2018\times 2020x=1$$

Calculate 2019 to the power of 2 and get 4076361.

$$4076361x^{2}+2018\times 2020x=1$$

Multiply 2018 and 2020 to get 4076360.

$$4076361x^{2}+4076360x=1$$

Divide both sides by 4076361.

$$\frac{4076361x^{2}+4076360x}{4076361}=\frac{1}{4076361}$$

Dividing by 4076361 undoes the multiplication by 4076361.

$$x^{2}+\frac{4076360}{4076361}x=\frac{1}{4076361}$$

Divide \frac{4076360}{4076361}, the coefficient of the x term, by 2 to get \frac{2038180}{4076361}. Then add the square of \frac{2038180}{4076361} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4076360}{4076361}x+\left(\frac{2038180}{4076361}\right)^{2}=\frac{1}{4076361}+\left(\frac{2038180}{4076361}\right)^{2}$$

Square \frac{2038180}{4076361} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4076360}{4076361}x+\frac{4154177712400}{16616719002321}=\frac{1}{4076361}+\frac{4154177712400}{16616719002321}$$

Add \frac{1}{4076361} to \frac{4154177712400}{16616719002321} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4076360}{4076361}x+\frac{4154177712400}{16616719002321}=\frac{4154181788761}{16616719002321}$$

Factor x^{2}+\frac{4076360}{4076361}x+\frac{4154177712400}{16616719002321}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2038180}{4076361}\right)^{2}=\frac{4154181788761}{16616719002321}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2038180}{4076361}\right)^{2}}=\sqrt{\frac{4154181788761}{16616719002321}}$$

Simplify.

$$x+\frac{2038180}{4076361}=\frac{2038181}{4076361} x+\frac{2038180}{4076361}=-\frac{2038181}{4076361}$$

Subtract \frac{2038180}{4076361} from both sides of the equation.

$$x=\frac{1}{4076361} x=-1$$