Solve 115=xleft(1+3x10^-3right)

$$115 = x \left( 1+3x { 10 }^{ -3 } \right)$$

Solve for x

$$x = \frac{50 \sqrt{238} - 500}{3} \approx 90.454143676$$

$$x=\frac{-50\sqrt{238}-500}{3}\approx -423.787477009$$

Steps Using the Quadratic Formula

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

$$115=x\left(1+3x\times \frac{1}{1000}\right)$$

Multiply 3 and \frac{1}{1000} to get \frac{3}{1000}.

$$115=x\left(1+\frac{3}{1000}x\right)$$

Use the distributive property to multiply x by 1+\frac{3}{1000}x.

$$115=x+\frac{3}{1000}x^{2}$$

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

$$x+\frac{3}{1000}x^{2}=115$$

Subtract 115 from both sides.

$$x+\frac{3}{1000}x^{2}-115=0$$

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.

$$\frac{3}{1000}x^{2}+x-115=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{1000} for a, 1 for b, and -115 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{3}{1000}\left(-115\right)}}{2\times \frac{3}{1000}}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times \frac{3}{1000}\left(-115\right)}}{2\times \frac{3}{1000}}$$

Multiply -4 times \frac{3}{1000}.

$$x=\frac{-1±\sqrt{1-\frac{3}{250}\left(-115\right)}}{2\times \frac{3}{1000}}$$

Multiply -\frac{3}{250} times -115.

$$x=\frac{-1±\sqrt{1+\frac{69}{50}}}{2\times \frac{3}{1000}}$$

Add 1 to \frac{69}{50}.

$$x=\frac{-1±\sqrt{\frac{119}{50}}}{2\times \frac{3}{1000}}$$

Take the square root of \frac{119}{50}.

$$x=\frac{-1±\frac{\sqrt{238}}{10}}{2\times \frac{3}{1000}}$$

Multiply 2 times \frac{3}{1000}.

$$x=\frac{-1±\frac{\sqrt{238}}{10}}{\frac{3}{500}}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{238}}{10}}{\frac{3}{500}} when ± is plus. Add -1 to \frac{\sqrt{238}}{10}.

$$x=\frac{\frac{\sqrt{238}}{10}-1}{\frac{3}{500}}$$

Divide -1+\frac{\sqrt{238}}{10} by \frac{3}{500} by multiplying -1+\frac{\sqrt{238}}{10} by the reciprocal of \frac{3}{500}.

$$x=\frac{50\sqrt{238}-500}{3}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{238}}{10}}{\frac{3}{500}} when ± is minus. Subtract \frac{\sqrt{238}}{10} from -1.

$$x=\frac{-\frac{\sqrt{238}}{10}-1}{\frac{3}{500}}$$

Divide -1-\frac{\sqrt{238}}{10} by \frac{3}{500} by multiplying -1-\frac{\sqrt{238}}{10} by the reciprocal of \frac{3}{500}.

$$x=\frac{-50\sqrt{238}-500}{3}$$

The equation is now solved.

$$x=\frac{50\sqrt{238}-500}{3} x=\frac{-50\sqrt{238}-500}{3}$$

Steps for Completing the Square

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

$$115=x\left(1+3x\times \frac{1}{1000}\right)$$

Multiply 3 and \frac{1}{1000} to get \frac{3}{1000}.

$$115=x\left(1+\frac{3}{1000}x\right)$$

Use the distributive property to multiply x by 1+\frac{3}{1000}x.

$$115=x+\frac{3}{1000}x^{2}$$

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

$$x+\frac{3}{1000}x^{2}=115$$

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.

$$\frac{3}{1000}x^{2}+x=115$$

Divide both sides of the equation by \frac{3}{1000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{1000}x^{2}+x}{\frac{3}{1000}}=\frac{115}{\frac{3}{1000}}$$

Dividing by \frac{3}{1000} undoes the multiplication by \frac{3}{1000}.

$$x^{2}+\frac{1}{\frac{3}{1000}}x=\frac{115}{\frac{3}{1000}}$$

Divide 1 by \frac{3}{1000} by multiplying 1 by the reciprocal of \frac{3}{1000}.

$$x^{2}+\frac{1000}{3}x=\frac{115}{\frac{3}{1000}}$$

Divide 115 by \frac{3}{1000} by multiplying 115 by the reciprocal of \frac{3}{1000}.

$$x^{2}+\frac{1000}{3}x=\frac{115000}{3}$$

Divide \frac{1000}{3}, the coefficient of the x term, by 2 to get \frac{500}{3}. Then add the square of \frac{500}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1000}{3}x+\left(\frac{500}{3}\right)^{2}=\frac{115000}{3}+\left(\frac{500}{3}\right)^{2}$$

Square \frac{500}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1000}{3}x+\frac{250000}{9}=\frac{115000}{3}+\frac{250000}{9}$$

Add \frac{115000}{3} to \frac{250000}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1000}{3}x+\frac{250000}{9}=\frac{595000}{9}$$

Factor x^{2}+\frac{1000}{3}x+\frac{250000}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{500}{3}\right)^{2}=\frac{595000}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{500}{3}\right)^{2}}=\sqrt{\frac{595000}{9}}$$

Simplify.

$$x+\frac{500}{3}=\frac{50\sqrt{238}}{3} x+\frac{500}{3}=-\frac{50\sqrt{238}}{3}$$

Subtract \frac{500}{3} from both sides of the equation.

$$x=\frac{50\sqrt{238}-500}{3} x=\frac{-50\sqrt{238}-500}{3}$$