Solve 4.63*10^-3(.0123-x)=x^2

$$4.63 \times 10 ^ { - 3 } ( .0123 - x ) = x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{2492329}-463}{200000}\approx 0.005578556$$

$$x=\frac{-\sqrt{2492329}-463}{200000}\approx -0.010208556$$

Steps Using the Quadratic Formula

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

$$4.63\times \frac{1}{1000}\left(0.0123-x\right)=x^{2}$$

Multiply 4.63 and \frac{1}{1000} to get \frac{463}{100000}.

$$\frac{463}{100000}\left(0.0123-x\right)=x^{2}$$

Use the distributive property to multiply \frac{463}{100000} by 0.0123-x.

$$\frac{56949}{1000000000}-\frac{463}{100000}x=x^{2}$$

Subtract x^{2} from both sides.

$$\frac{56949}{1000000000}-\frac{463}{100000}x-x^{2}=0$$

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.

$$-x^{2}-\frac{463}{100000}x+\frac{56949}{1000000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{463}{100000} for b, and \frac{56949}{1000000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\sqrt{\left(-\frac{463}{100000}\right)^{2}-4\left(-1\right)\times \frac{56949}{1000000000}}}{2\left(-1\right)}$$

Square -\frac{463}{100000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\sqrt{\frac{214369}{10000000000}-4\left(-1\right)\times \frac{56949}{1000000000}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\sqrt{\frac{214369}{10000000000}+4\times \frac{56949}{1000000000}}}{2\left(-1\right)}$$

Multiply 4 times \frac{56949}{1000000000}.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\sqrt{\frac{214369}{10000000000}+\frac{56949}{250000000}}}{2\left(-1\right)}$$

Add \frac{214369}{10000000000} to \frac{56949}{250000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\sqrt{\frac{2492329}{10000000000}}}{2\left(-1\right)}$$

Take the square root of \frac{2492329}{10000000000}.

$$x=\frac{-\left(-\frac{463}{100000}\right)±\frac{\sqrt{2492329}}{100000}}{2\left(-1\right)}$$

The opposite of -\frac{463}{100000} is \frac{463}{100000}.

$$x=\frac{\frac{463}{100000}±\frac{\sqrt{2492329}}{100000}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{463}{100000}±\frac{\sqrt{2492329}}{100000}}{-2}$$

Now solve the equation x=\frac{\frac{463}{100000}±\frac{\sqrt{2492329}}{100000}}{-2} when ± is plus. Add \frac{463}{100000} to \frac{\sqrt{2492329}}{100000}.

$$x=\frac{\sqrt{2492329}+463}{-2\times 100000}$$

Divide \frac{463+\sqrt{2492329}}{100000} by -2.

$$x=\frac{-\sqrt{2492329}-463}{200000}$$

Now solve the equation x=\frac{\frac{463}{100000}±\frac{\sqrt{2492329}}{100000}}{-2} when ± is minus. Subtract \frac{\sqrt{2492329}}{100000} from \frac{463}{100000}.

$$x=\frac{463-\sqrt{2492329}}{-2\times 100000}$$

Divide \frac{463-\sqrt{2492329}}{100000} by -2.

$$x=\frac{\sqrt{2492329}-463}{200000}$$

The equation is now solved.

$$x=\frac{-\sqrt{2492329}-463}{200000} x=\frac{\sqrt{2492329}-463}{200000}$$

Steps for Completing the Square

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

$$4.63\times \frac{1}{1000}\left(0.0123-x\right)=x^{2}$$

Multiply 4.63 and \frac{1}{1000} to get \frac{463}{100000}.

$$\frac{463}{100000}\left(0.0123-x\right)=x^{2}$$

Use the distributive property to multiply \frac{463}{100000} by 0.0123-x.

$$\frac{56949}{1000000000}-\frac{463}{100000}x=x^{2}$$

Subtract x^{2} from both sides.

$$\frac{56949}{1000000000}-\frac{463}{100000}x-x^{2}=0$$

Subtract \frac{56949}{1000000000} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{463}{100000}x-x^{2}=-\frac{56949}{1000000000}$$

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.

$$-x^{2}-\frac{463}{100000}x=-\frac{56949}{1000000000}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{463}{100000}x}{-1}=-\frac{\frac{56949}{1000000000}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{463}{100000}}{-1}\right)x=-\frac{\frac{56949}{1000000000}}{-1}$$

Divide -\frac{463}{100000} by -1.

$$x^{2}+\frac{463}{100000}x=-\frac{\frac{56949}{1000000000}}{-1}$$

Divide -\frac{56949}{1000000000} by -1.

$$x^{2}+\frac{463}{100000}x=\frac{56949}{1000000000}$$

Divide \frac{463}{100000}, the coefficient of the x term, by 2 to get \frac{463}{200000}. Then add the square of \frac{463}{200000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{463}{100000}x+\left(\frac{463}{200000}\right)^{2}=\frac{56949}{1000000000}+\left(\frac{463}{200000}\right)^{2}$$

Square \frac{463}{200000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{463}{100000}x+\frac{214369}{40000000000}=\frac{56949}{1000000000}+\frac{214369}{40000000000}$$

Add \frac{56949}{1000000000} to \frac{214369}{40000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{463}{100000}x+\frac{214369}{40000000000}=\frac{2492329}{40000000000}$$

Factor x^{2}+\frac{463}{100000}x+\frac{214369}{40000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{463}{200000}\right)^{2}=\frac{2492329}{40000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{463}{200000}\right)^{2}}=\sqrt{\frac{2492329}{40000000000}}$$

Simplify.

$$x+\frac{463}{200000}=\frac{\sqrt{2492329}}{200000} x+\frac{463}{200000}=-\frac{\sqrt{2492329}}{200000}$$

Subtract \frac{463}{200000} from both sides of the equation.

$$x=\frac{\sqrt{2492329}-463}{200000} x=\frac{-\sqrt{2492329}-463}{200000}$$