Solve 463*10^-3(0123-x)=x^2

$$463 \times 10 ^ { - 3 } ( 0123 - x ) = x ^ { 2 }$$

Solve for x

$$x=-\frac{463}{1000}=-0.463$$

$$x=0$$

Steps Using Factoring

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

$$463\times \frac{1}{1000}\left(0\times 123-x\right)=x^{2}$$

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

$$\frac{463}{1000}\left(0\times 123-x\right)=x^{2}$$

Multiply 0 and 123 to get 0.

$$\frac{463}{1000}\left(0-x\right)=x^{2}$$

Anything plus zero gives itself.

$$\frac{463}{1000}\left(-1\right)x=x^{2}$$

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

$$-\frac{463}{1000}x=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{463}{1000}x-x^{2}=0$$

Factor out x.

$$x\left(-\frac{463}{1000}-x\right)=0$$

To find equation solutions, solve x=0 and -\frac{463}{1000}-x=0.

$$x=0 x=-\frac{463}{1000}$$

Steps Using the Quadratic Formula

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

$$463\times \frac{1}{1000}\left(0\times 123-x\right)=x^{2}$$

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

$$\frac{463}{1000}\left(0\times 123-x\right)=x^{2}$$

Multiply 0 and 123 to get 0.

$$\frac{463}{1000}\left(0-x\right)=x^{2}$$

Anything plus zero gives itself.

$$\frac{463}{1000}\left(-1\right)x=x^{2}$$

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

$$-\frac{463}{1000}x=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{463}{1000}x-x^{2}=0$$

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.

$$-x^{2}-\frac{463}{1000}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{463}{1000} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{463}{1000}\right)±\sqrt{\left(-\frac{463}{1000}\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(-\frac{463}{1000}\right)^{2}.

$$x=\frac{-\left(-\frac{463}{1000}\right)±\frac{463}{1000}}{2\left(-1\right)}$$

The opposite of -\frac{463}{1000} is \frac{463}{1000}.

$$x=\frac{\frac{463}{1000}±\frac{463}{1000}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{463}{1000}±\frac{463}{1000}}{-2}$$

Now solve the equation x=\frac{\frac{463}{1000}±\frac{463}{1000}}{-2} when ± is plus. Add \frac{463}{1000} to \frac{463}{1000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{463}{500}}{-2}$$

Divide \frac{463}{500} by -2.

$$x=-\frac{463}{1000}$$

Now solve the equation x=\frac{\frac{463}{1000}±\frac{463}{1000}}{-2} when ± is minus. Subtract \frac{463}{1000} from \frac{463}{1000} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

The equation is now solved.

$$x=-\frac{463}{1000} x=0$$

Steps for Completing the Square

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

$$463\times \frac{1}{1000}\left(0\times 123-x\right)=x^{2}$$

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

$$\frac{463}{1000}\left(0\times 123-x\right)=x^{2}$$

Multiply 0 and 123 to get 0.

$$\frac{463}{1000}\left(0-x\right)=x^{2}$$

Anything plus zero gives itself.

$$\frac{463}{1000}\left(-1\right)x=x^{2}$$

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

$$-\frac{463}{1000}x=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{463}{1000}x-x^{2}=0$$

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.

$$-x^{2}-\frac{463}{1000}x=0$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{463}{1000}x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{463}{1000}}{-1}\right)x=\frac{0}{-1}$$

Divide -\frac{463}{1000} by -1.

$$x^{2}+\frac{463}{1000}x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}+\frac{463}{1000}x=0$$

Divide \frac{463}{1000}, the coefficient of the x term, by 2 to get \frac{463}{2000}. Then add the square of \frac{463}{2000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{463}{1000}x+\left(\frac{463}{2000}\right)^{2}=\left(\frac{463}{2000}\right)^{2}$$

Square \frac{463}{2000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{463}{1000}x+\frac{214369}{4000000}=\frac{214369}{4000000}$$

Factor x^{2}+\frac{463}{1000}x+\frac{214369}{4000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{463}{2000}\right)^{2}=\frac{214369}{4000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{463}{2000}\right)^{2}}=\sqrt{\frac{214369}{4000000}}$$

Simplify.

$$x+\frac{463}{2000}=\frac{463}{2000} x+\frac{463}{2000}=-\frac{463}{2000}$$

Subtract \frac{463}{2000} from both sides of the equation.

$$x=0 x=-\frac{463}{1000}$$