Solve 0.0007x^2-0.017x-0.73=0

$$0.0007 { x }^{ 2 } -0.017x-0.73=0$$

Solve for x

$$x = \frac{5 \sqrt{2333} + 85}{7} \approx 46.643670502$$

$$x=\frac{85-5\sqrt{2333}}{7}\approx -22.357956216$$

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.

$$0.0007x^{2}-0.017x-0.73=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0007 for a, -0.017 for b, and -0.73 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.017\right)±\sqrt{\left(-0.017\right)^{2}-4\times 0.0007\left(-0.73\right)}}{2\times 0.0007}$$

Square -0.017 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.017\right)±\sqrt{0.000289-4\times 0.0007\left(-0.73\right)}}{2\times 0.0007}$$

Multiply -4 times 0.0007.

$$x=\frac{-\left(-0.017\right)±\sqrt{0.000289-0.0028\left(-0.73\right)}}{2\times 0.0007}$$

Multiply -0.0028 times -0.73 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.017\right)±\sqrt{0.000289+0.002044}}{2\times 0.0007}$$

Add 0.000289 to 0.002044 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.017\right)±\sqrt{0.002333}}{2\times 0.0007}$$

Take the square root of 0.002333.

$$x=\frac{-\left(-0.017\right)±\frac{\sqrt{2333}}{1000}}{2\times 0.0007}$$

The opposite of -0.017 is 0.017.

$$x=\frac{0.017±\frac{\sqrt{2333}}{1000}}{2\times 0.0007}$$

Multiply 2 times 0.0007.

$$x=\frac{0.017±\frac{\sqrt{2333}}{1000}}{0.0014}$$

Now solve the equation x=\frac{0.017±\frac{\sqrt{2333}}{1000}}{0.0014} when ± is plus. Add 0.017 to \frac{\sqrt{2333}}{1000}.

$$x=\frac{\sqrt{2333}+17}{0.0014\times 1000}$$

Divide \frac{17+\sqrt{2333}}{1000} by 0.0014 by multiplying \frac{17+\sqrt{2333}}{1000} by the reciprocal of 0.0014.

$$x=\frac{5\sqrt{2333}+85}{7}$$

Now solve the equation x=\frac{0.017±\frac{\sqrt{2333}}{1000}}{0.0014} when ± is minus. Subtract \frac{\sqrt{2333}}{1000} from 0.017.

$$x=\frac{17-\sqrt{2333}}{0.0014\times 1000}$$

Divide \frac{17-\sqrt{2333}}{1000} by 0.0014 by multiplying \frac{17-\sqrt{2333}}{1000} by the reciprocal of 0.0014.

$$x=\frac{85-5\sqrt{2333}}{7}$$

The equation is now solved.

$$x=\frac{5\sqrt{2333}+85}{7} x=\frac{85-5\sqrt{2333}}{7}$$

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.

$$0.0007x^{2}-0.017x-0.73=0$$

Add 0.73 to both sides of the equation.

$$0.0007x^{2}-0.017x-0.73-\left(-0.73\right)=-\left(-0.73\right)$$

Subtracting -0.73 from itself leaves 0.

$$0.0007x^{2}-0.017x=-\left(-0.73\right)$$

Subtract -0.73 from 0.

$$0.0007x^{2}-0.017x=0.73$$

Divide both sides of the equation by 0.0007, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0007x^{2}-0.017x}{0.0007}=\frac{0.73}{0.0007}$$

Dividing by 0.0007 undoes the multiplication by 0.0007.

$$x^{2}+\left(-\frac{0.017}{0.0007}\right)x=\frac{0.73}{0.0007}$$

Divide -0.017 by 0.0007 by multiplying -0.017 by the reciprocal of 0.0007.

$$x^{2}-\frac{170}{7}x=\frac{0.73}{0.0007}$$

Divide 0.73 by 0.0007 by multiplying 0.73 by the reciprocal of 0.0007.

$$x^{2}-\frac{170}{7}x=\frac{7300}{7}$$

Divide -\frac{170}{7}, the coefficient of the x term, by 2 to get -\frac{85}{7}. Then add the square of -\frac{85}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{170}{7}x+\left(-\frac{85}{7}\right)^{2}=\frac{7300}{7}+\left(-\frac{85}{7}\right)^{2}$$

Square -\frac{85}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{170}{7}x+\frac{7225}{49}=\frac{7300}{7}+\frac{7225}{49}$$

Add \frac{7300}{7} to \frac{7225}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{170}{7}x+\frac{7225}{49}=\frac{58325}{49}$$

Factor x^{2}-\frac{170}{7}x+\frac{7225}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{85}{7}\right)^{2}=\frac{58325}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{7}\right)^{2}}=\sqrt{\frac{58325}{49}}$$

Simplify.

$$x-\frac{85}{7}=\frac{5\sqrt{2333}}{7} x-\frac{85}{7}=-\frac{5\sqrt{2333}}{7}$$

Add \frac{85}{7} to both sides of the equation.

$$x=\frac{5\sqrt{2333}+85}{7} x=\frac{85-5\sqrt{2333}}{7}$$