Solve x^2+(1+(1.69*10^-3))x-(1.69*10^-3)=0

$${ x }^{ 2 } +(1+(1.69 \times { 10 }^{ -3 } ))x-(1.69 \times { 10 }^{ -3 } )=0$$

Solve for x

$$x=\frac{\sqrt{10101428561}-100169}{200000}\approx 0.001684317$$

$$x=\frac{-\sqrt{10101428561}-100169}{200000}\approx -1.003374317$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+\left(1+1.69\times \frac{1}{1000}\right)x-1.69\times 10^{-3}=0$$

Multiply 1.69 and \frac{1}{1000} to get \frac{169}{100000}.

$$x^{2}+\left(1+\frac{169}{100000}\right)x-1.69\times 10^{-3}=0$$

Add 1 and \frac{169}{100000} to get \frac{100169}{100000}.

$$x^{2}+\frac{100169}{100000}x-1.69\times 10^{-3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+\frac{100169}{100000}x-1.69\times \frac{1}{1000}=0$$

Multiply 1.69 and \frac{1}{1000} to get \frac{169}{100000}.

$$x^{2}+\frac{100169}{100000}x-\frac{169}{100000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{100169}{100000} for b, and -\frac{169}{100000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{100169}{100000}±\sqrt{\left(\frac{100169}{100000}\right)^{2}-4\left(-\frac{169}{100000}\right)}}{2}$$

Square \frac{100169}{100000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{100169}{100000}±\sqrt{\frac{10033828561}{10000000000}-4\left(-\frac{169}{100000}\right)}}{2}$$

Multiply -4 times -\frac{169}{100000}.

$$x=\frac{-\frac{100169}{100000}±\sqrt{\frac{10033828561}{10000000000}+\frac{169}{25000}}}{2}$$

Add \frac{10033828561}{10000000000} to \frac{169}{25000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{100169}{100000}±\sqrt{\frac{10101428561}{10000000000}}}{2}$$

Take the square root of \frac{10101428561}{10000000000}.

$$x=\frac{-\frac{100169}{100000}±\frac{\sqrt{10101428561}}{100000}}{2}$$

Now solve the equation x=\frac{-\frac{100169}{100000}±\frac{\sqrt{10101428561}}{100000}}{2} when ± is plus. Add -\frac{100169}{100000} to \frac{\sqrt{10101428561}}{100000}.

$$x=\frac{\sqrt{10101428561}-100169}{2\times 100000}$$

Divide \frac{-100169+\sqrt{10101428561}}{100000} by 2.

$$x=\frac{\sqrt{10101428561}-100169}{200000}$$

Now solve the equation x=\frac{-\frac{100169}{100000}±\frac{\sqrt{10101428561}}{100000}}{2} when ± is minus. Subtract \frac{\sqrt{10101428561}}{100000} from -\frac{100169}{100000}.

$$x=\frac{-\sqrt{10101428561}-100169}{2\times 100000}$$

Divide \frac{-100169-\sqrt{10101428561}}{100000} by 2.

$$x=\frac{-\sqrt{10101428561}-100169}{200000}$$

The equation is now solved.

$$x=\frac{\sqrt{10101428561}-100169}{200000} x=\frac{-\sqrt{10101428561}-100169}{200000}$$

Steps for Completing the Square

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+\left(1+1.69\times \frac{1}{1000}\right)x-1.69\times 10^{-3}=0$$

Multiply 1.69 and \frac{1}{1000} to get \frac{169}{100000}.

$$x^{2}+\left(1+\frac{169}{100000}\right)x-1.69\times 10^{-3}=0$$

Add 1 and \frac{169}{100000} to get \frac{100169}{100000}.

$$x^{2}+\frac{100169}{100000}x-1.69\times 10^{-3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+\frac{100169}{100000}x-1.69\times \frac{1}{1000}=0$$

Multiply 1.69 and \frac{1}{1000} to get \frac{169}{100000}.

$$x^{2}+\frac{100169}{100000}x-\frac{169}{100000}=0$$

Add \frac{169}{100000} to both sides. Anything plus zero gives itself.

$$x^{2}+\frac{100169}{100000}x=\frac{169}{100000}$$

Divide \frac{100169}{100000}, the coefficient of the x term, by 2 to get \frac{100169}{200000}. Then add the square of \frac{100169}{200000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{100169}{100000}x+\left(\frac{100169}{200000}\right)^{2}=\frac{169}{100000}+\left(\frac{100169}{200000}\right)^{2}$$

Square \frac{100169}{200000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{100169}{100000}x+\frac{10033828561}{40000000000}=\frac{169}{100000}+\frac{10033828561}{40000000000}$$

Add \frac{169}{100000} to \frac{10033828561}{40000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{100169}{100000}x+\frac{10033828561}{40000000000}=\frac{10101428561}{40000000000}$$

Factor x^{2}+\frac{100169}{100000}x+\frac{10033828561}{40000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{100169}{200000}\right)^{2}=\frac{10101428561}{40000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{100169}{200000}\right)^{2}}=\sqrt{\frac{10101428561}{40000000000}}$$

Simplify.

$$x+\frac{100169}{200000}=\frac{\sqrt{10101428561}}{200000} x+\frac{100169}{200000}=-\frac{\sqrt{10101428561}}{200000}$$

Subtract \frac{100169}{200000} from both sides of the equation.

$$x=\frac{\sqrt{10101428561}-100169}{200000} x=\frac{-\sqrt{10101428561}-100169}{200000}$$