Solve 0.4=0.0017x^2+0.0739x+0.0014

$$0.4 = 0.0017 x ^ { 2 } + 0.0739 x + 0.0014$$

Solve for x

$$x = \frac{\sqrt{817169} - 739}{34} \approx 4.85217659$$

$$x=\frac{-\sqrt{817169}-739}{34}\approx -48.322764825$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$0.0017x^{2}+0.0739x+0.0014=0.4$$

Subtract 0.4 from both sides.

$$0.0017x^{2}+0.0739x+0.0014-0.4=0$$

Subtract 0.4 from 0.0014 to get -0.3986.

$$0.0017x^{2}+0.0739x-0.3986=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0017 for a, 0.0739 for b, and -0.3986 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.0739±\sqrt{0.0739^{2}-4\times 0.0017\left(-0.3986\right)}}{2\times 0.0017}$$

Square 0.0739 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.0739±\sqrt{0.00546121-4\times 0.0017\left(-0.3986\right)}}{2\times 0.0017}$$

Multiply -4 times 0.0017.

$$x=\frac{-0.0739±\sqrt{0.00546121-0.0068\left(-0.3986\right)}}{2\times 0.0017}$$

Multiply -0.0068 times -0.3986 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0739±\sqrt{0.00546121+0.00271048}}{2\times 0.0017}$$

Add 0.00546121 to 0.00271048 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0739±\sqrt{0.00817169}}{2\times 0.0017}$$

Take the square root of 0.00817169.

$$x=\frac{-0.0739±\frac{\sqrt{817169}}{10000}}{2\times 0.0017}$$

Multiply 2 times 0.0017.

$$x=\frac{-0.0739±\frac{\sqrt{817169}}{10000}}{0.0034}$$

Now solve the equation x=\frac{-0.0739±\frac{\sqrt{817169}}{10000}}{0.0034} when ± is plus. Add -0.0739 to \frac{\sqrt{817169}}{10000}.

$$x=\frac{\sqrt{817169}-739}{0.0034\times 10000}$$

Divide \frac{-739+\sqrt{817169}}{10000} by 0.0034 by multiplying \frac{-739+\sqrt{817169}}{10000} by the reciprocal of 0.0034.

$$x=\frac{\sqrt{817169}-739}{34}$$

Now solve the equation x=\frac{-0.0739±\frac{\sqrt{817169}}{10000}}{0.0034} when ± is minus. Subtract \frac{\sqrt{817169}}{10000} from -0.0739.

$$x=\frac{-\sqrt{817169}-739}{0.0034\times 10000}$$

Divide \frac{-739-\sqrt{817169}}{10000} by 0.0034 by multiplying \frac{-739-\sqrt{817169}}{10000} by the reciprocal of 0.0034.

$$x=\frac{-\sqrt{817169}-739}{34}$$

The equation is now solved.

$$x=\frac{\sqrt{817169}-739}{34} x=\frac{-\sqrt{817169}-739}{34}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$0.0017x^{2}+0.0739x+0.0014=0.4$$

Subtract 0.0014 from both sides.

$$0.0017x^{2}+0.0739x=0.4-0.0014$$

Subtract 0.0014 from 0.4 to get 0.3986.

$$0.0017x^{2}+0.0739x=0.3986$$

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.

$$0.0017x^{2}+0.0739x=\frac{1993}{5000}$$

Divide both sides of the equation by 0.0017, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0017x^{2}+0.0739x}{0.0017}=\frac{\frac{1993}{5000}}{0.0017}$$

Dividing by 0.0017 undoes the multiplication by 0.0017.

$$x^{2}+\frac{0.0739}{0.0017}x=\frac{\frac{1993}{5000}}{0.0017}$$

Divide 0.0739 by 0.0017 by multiplying 0.0739 by the reciprocal of 0.0017.

$$x^{2}+\frac{739}{17}x=\frac{\frac{1993}{5000}}{0.0017}$$

Divide \frac{1993}{5000} by 0.0017 by multiplying \frac{1993}{5000} by the reciprocal of 0.0017.

$$x^{2}+\frac{739}{17}x=\frac{3986}{17}$$

Divide \frac{739}{17}, the coefficient of the x term, by 2 to get \frac{739}{34}. Then add the square of \frac{739}{34} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{739}{17}x+\frac{739}{34}^{2}=\frac{3986}{17}+\frac{739}{34}^{2}$$

Square \frac{739}{34} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{739}{17}x+\frac{546121}{1156}=\frac{3986}{17}+\frac{546121}{1156}$$

Add \frac{3986}{17} to \frac{546121}{1156} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{739}{17}x+\frac{546121}{1156}=\frac{817169}{1156}$$

Factor x^{2}+\frac{739}{17}x+\frac{546121}{1156}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{739}{34}\right)^{2}=\frac{817169}{1156}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{739}{34}\right)^{2}}=\sqrt{\frac{817169}{1156}}$$

Simplify.

$$x+\frac{739}{34}=\frac{\sqrt{817169}}{34} x+\frac{739}{34}=-\frac{\sqrt{817169}}{34}$$

Subtract \frac{739}{34} from both sides of the equation.

$$x=\frac{\sqrt{817169}-739}{34} x=\frac{-\sqrt{817169}-739}{34}$$