Solve 0.0005x^2+0.0539x-1.5816=1.082

$$0.0005 { x }^{ 2 } +0.0539x-1.5816 = 1.082$$

Solve for x

$$x = \frac{\sqrt{823241} - 539}{10} \approx 36.832629191$$

$$x=\frac{-\sqrt{823241}-539}{10}\approx -144.632629191$$

Steps Using the Quadratic Formula

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.

$$0.0005x^{2}+0.0539x-1.5816=1.082$$

Subtract 1.082 from both sides of the equation.

$$0.0005x^{2}+0.0539x-1.5816-1.082=1.082-1.082$$

Subtracting 1.082 from itself leaves 0.

$$0.0005x^{2}+0.0539x-1.5816-1.082=0$$

Subtract 1.082 from -1.5816 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$0.0005x^{2}+0.0539x-2.6636=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0005 for a, 0.0539 for b, and -2.6636 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.0539±\sqrt{0.0539^{2}-4\times 0.0005\left(-2.6636\right)}}{2\times 0.0005}$$

Square 0.0539 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.0539±\sqrt{0.00290521-4\times 0.0005\left(-2.6636\right)}}{2\times 0.0005}$$

Multiply -4 times 0.0005.

$$x=\frac{-0.0539±\sqrt{0.00290521-0.002\left(-2.6636\right)}}{2\times 0.0005}$$

Multiply -0.002 times -2.6636 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0539±\sqrt{0.00290521+0.0053272}}{2\times 0.0005}$$

Add 0.00290521 to 0.0053272 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0539±\sqrt{0.00823241}}{2\times 0.0005}$$

Take the square root of 0.00823241.

$$x=\frac{-0.0539±\frac{\sqrt{823241}}{10000}}{2\times 0.0005}$$

Multiply 2 times 0.0005.

$$x=\frac{-0.0539±\frac{\sqrt{823241}}{10000}}{0.001}$$

Now solve the equation x=\frac{-0.0539±\frac{\sqrt{823241}}{10000}}{0.001} when ± is plus. Add -0.0539 to \frac{\sqrt{823241}}{10000}.

$$x=\frac{\sqrt{823241}-539}{0.001\times 10000}$$

Divide \frac{-539+\sqrt{823241}}{10000} by 0.001 by multiplying \frac{-539+\sqrt{823241}}{10000} by the reciprocal of 0.001.

$$x=\frac{\sqrt{823241}-539}{10}$$

Now solve the equation x=\frac{-0.0539±\frac{\sqrt{823241}}{10000}}{0.001} when ± is minus. Subtract \frac{\sqrt{823241}}{10000} from -0.0539.

$$x=\frac{-\sqrt{823241}-539}{0.001\times 10000}$$

Divide \frac{-539-\sqrt{823241}}{10000} by 0.001 by multiplying \frac{-539-\sqrt{823241}}{10000} by the reciprocal of 0.001.

$$x=\frac{-\sqrt{823241}-539}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{823241}-539}{10} x=\frac{-\sqrt{823241}-539}{10}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.0005x^{2}+0.0539x-1.5816=1.082$$

Add 1.5816 to both sides of the equation.

$$0.0005x^{2}+0.0539x-1.5816-\left(-1.5816\right)=1.082-\left(-1.5816\right)$$

Subtracting -1.5816 from itself leaves 0.

$$0.0005x^{2}+0.0539x=1.082-\left(-1.5816\right)$$

Subtract -1.5816 from 1.082 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$0.0005x^{2}+0.0539x=2.6636$$

Multiply both sides by 2000.

$$\frac{0.0005x^{2}+0.0539x}{0.0005}=\frac{2.6636}{0.0005}$$

Dividing by 0.0005 undoes the multiplication by 0.0005.

$$x^{2}+\frac{0.0539}{0.0005}x=\frac{2.6636}{0.0005}$$

Divide 0.0539 by 0.0005 by multiplying 0.0539 by the reciprocal of 0.0005.

$$x^{2}+107.8x=\frac{2.6636}{0.0005}$$

Divide 2.6636 by 0.0005 by multiplying 2.6636 by the reciprocal of 0.0005.

$$x^{2}+107.8x=5327.2$$

Divide 107.8, the coefficient of the x term, by 2 to get 53.9. Then add the square of 53.9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+107.8x+53.9^{2}=5327.2+53.9^{2}$$

Square 53.9 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+107.8x+2905.21=5327.2+2905.21$$

Add 5327.2 to 2905.21 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+107.8x+2905.21=8232.41$$

Factor x^{2}+107.8x+2905.21. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+53.9\right)^{2}=8232.41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+53.9\right)^{2}}=\sqrt{8232.41}$$

Simplify.

$$x+53.9=\frac{\sqrt{823241}}{10} x+53.9=-\frac{\sqrt{823241}}{10}$$

Subtract 53.9 from both sides of the equation.

$$x=\frac{\sqrt{823241}-539}{10} x=\frac{-\sqrt{823241}-539}{10}$$