Solve .82*10^-5=[1.48*10^-3+x][0.0107+x]/7

$$.82 \times 10 ^ { - 5 } = \frac { [ 1.48 \times 10 ^ { - 3 } + x ] [ 0.0107 + x ] } { 7 }$$

Solve for x

$$x=\frac{\sqrt{786521}-609}{100000}\approx 0.002778602$$

$$x=\frac{-\sqrt{786521}-609}{100000}\approx -0.014958602$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 7.

$$5.74\times 10^{-5}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$5.74\times \frac{1}{100000}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

Multiply 5.74 and \frac{1}{100000} to get \frac{287}{5000000}.

$$\frac{287}{5000000}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

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

$$\frac{287}{5000000}=\left(1.48\times \frac{1}{1000}+x\right)\left(0.0107+x\right)$$

Multiply 1.48 and \frac{1}{1000} to get \frac{37}{25000}.

$$\frac{287}{5000000}=\left(\frac{37}{25000}+x\right)\left(0.0107+x\right)$$

Use the distributive property to multiply \frac{37}{25000}+x by 0.0107+x and combine like terms.

$$\frac{287}{5000000}=\frac{3959}{250000000}+\frac{609}{50000}x+x^{2}$$

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

$$\frac{3959}{250000000}+\frac{609}{50000}x+x^{2}=\frac{287}{5000000}$$

Subtract \frac{287}{5000000} from both sides.

$$\frac{3959}{250000000}+\frac{609}{50000}x+x^{2}-\frac{287}{5000000}=0$$

Subtract \frac{287}{5000000} from \frac{3959}{250000000} to get -\frac{10391}{250000000}.

$$-\frac{10391}{250000000}+\frac{609}{50000}x+x^{2}=0$$

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.

$$x^{2}+\frac{609}{50000}x-\frac{10391}{250000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{609}{50000} for b, and -\frac{10391}{250000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{609}{50000}±\sqrt{\left(\frac{609}{50000}\right)^{2}-4\left(-\frac{10391}{250000000}\right)}}{2}$$

Square \frac{609}{50000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{609}{50000}±\sqrt{\frac{370881}{2500000000}-4\left(-\frac{10391}{250000000}\right)}}{2}$$

Multiply -4 times -\frac{10391}{250000000}.

$$x=\frac{-\frac{609}{50000}±\sqrt{\frac{370881}{2500000000}+\frac{10391}{62500000}}}{2}$$

Add \frac{370881}{2500000000} to \frac{10391}{62500000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{609}{50000}±\sqrt{\frac{786521}{2500000000}}}{2}$$

Take the square root of \frac{786521}{2500000000}.

$$x=\frac{-\frac{609}{50000}±\frac{\sqrt{786521}}{50000}}{2}$$

Now solve the equation x=\frac{-\frac{609}{50000}±\frac{\sqrt{786521}}{50000}}{2} when ± is plus. Add -\frac{609}{50000} to \frac{\sqrt{786521}}{50000}.

$$x=\frac{\sqrt{786521}-609}{2\times 50000}$$

Divide \frac{-609+\sqrt{786521}}{50000} by 2.

$$x=\frac{\sqrt{786521}-609}{100000}$$

Now solve the equation x=\frac{-\frac{609}{50000}±\frac{\sqrt{786521}}{50000}}{2} when ± is minus. Subtract \frac{\sqrt{786521}}{50000} from -\frac{609}{50000}.

$$x=\frac{-\sqrt{786521}-609}{2\times 50000}$$

Divide \frac{-609-\sqrt{786521}}{50000} by 2.

$$x=\frac{-\sqrt{786521}-609}{100000}$$

The equation is now solved.

$$x=\frac{\sqrt{786521}-609}{100000} x=\frac{-\sqrt{786521}-609}{100000}$$

Steps for Completing the Square

Multiply both sides of the equation by 7.

$$5.74\times 10^{-5}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$5.74\times \frac{1}{100000}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

Multiply 5.74 and \frac{1}{100000} to get \frac{287}{5000000}.

$$\frac{287}{5000000}=\left(1.48\times 10^{-3}+x\right)\left(0.0107+x\right)$$

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

$$\frac{287}{5000000}=\left(1.48\times \frac{1}{1000}+x\right)\left(0.0107+x\right)$$

Multiply 1.48 and \frac{1}{1000} to get \frac{37}{25000}.

$$\frac{287}{5000000}=\left(\frac{37}{25000}+x\right)\left(0.0107+x\right)$$

Use the distributive property to multiply \frac{37}{25000}+x by 0.0107+x and combine like terms.

$$\frac{287}{5000000}=\frac{3959}{250000000}+\frac{609}{50000}x+x^{2}$$

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

$$\frac{3959}{250000000}+\frac{609}{50000}x+x^{2}=\frac{287}{5000000}$$

Subtract \frac{3959}{250000000} from both sides.

$$\frac{609}{50000}x+x^{2}=\frac{287}{5000000}-\frac{3959}{250000000}$$

Subtract \frac{3959}{250000000} from \frac{287}{5000000} to get \frac{10391}{250000000}.

$$\frac{609}{50000}x+x^{2}=\frac{10391}{250000000}$$

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.

$$x^{2}+\frac{609}{50000}x=\frac{10391}{250000000}$$

Divide \frac{609}{50000}, the coefficient of the x term, by 2 to get \frac{609}{100000}. Then add the square of \frac{609}{100000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{609}{50000}x+\left(\frac{609}{100000}\right)^{2}=\frac{10391}{250000000}+\left(\frac{609}{100000}\right)^{2}$$

Square \frac{609}{100000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{609}{50000}x+\frac{370881}{10000000000}=\frac{10391}{250000000}+\frac{370881}{10000000000}$$

Add \frac{10391}{250000000} to \frac{370881}{10000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{609}{50000}x+\frac{370881}{10000000000}=\frac{786521}{10000000000}$$

Factor x^{2}+\frac{609}{50000}x+\frac{370881}{10000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{609}{100000}\right)^{2}=\frac{786521}{10000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{609}{100000}\right)^{2}}=\sqrt{\frac{786521}{10000000000}}$$

Simplify.

$$x+\frac{609}{100000}=\frac{\sqrt{786521}}{100000} x+\frac{609}{100000}=-\frac{\sqrt{786521}}{100000}$$

Subtract \frac{609}{100000} from both sides of the equation.

$$x=\frac{\sqrt{786521}-609}{100000} x=\frac{-\sqrt{786521}-609}{100000}$$