Solve 5-x/4*10^6=9.6x-x^2

$$\frac { 5 - x } { 4 \times 10 ^ { 6 } } = 9.6 x - x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{1474559996800001} + 38400001}{8000000} \approx 9.60000012$$

$$x=\frac{38400001-\sqrt{1474559996800001}}{8000000}\approx 0.00000013$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 6 and get 1000000.

$$\frac{5-x}{4\times 1000000}=9.6x-x^{2}$$

Multiply 4 and 1000000 to get 4000000.

$$\frac{5-x}{4000000}=9.6x-x^{2}$$

Divide each term of 5-x by 4000000 to get \frac{1}{800000}-\frac{1}{4000000}x.

$$\frac{1}{800000}-\frac{1}{4000000}x=9.6x-x^{2}$$

Subtract 9.6x from both sides.

$$\frac{1}{800000}-\frac{1}{4000000}x-9.6x=-x^{2}$$

Combine -\frac{1}{4000000}x and -9.6x to get -\frac{38400001}{4000000}x.

$$\frac{1}{800000}-\frac{38400001}{4000000}x=-x^{2}$$

Add x^{2} to both sides.

$$\frac{1}{800000}-\frac{38400001}{4000000}x+x^{2}=0$$

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.

$$x^{2}-\frac{38400001}{4000000}x+\frac{1}{800000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{38400001}{4000000} for b, and \frac{1}{800000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{38400001}{4000000}\right)±\sqrt{\left(-\frac{38400001}{4000000}\right)^{2}-4\times \frac{1}{800000}}}{2}$$

Square -\frac{38400001}{4000000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{38400001}{4000000}\right)±\sqrt{\frac{1474560076800001}{16000000000000}-4\times \frac{1}{800000}}}{2}$$

Multiply -4 times \frac{1}{800000}.

$$x=\frac{-\left(-\frac{38400001}{4000000}\right)±\sqrt{\frac{1474560076800001}{16000000000000}-\frac{1}{200000}}}{2}$$

Add \frac{1474560076800001}{16000000000000} to -\frac{1}{200000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{38400001}{4000000}\right)±\sqrt{\frac{1474559996800001}{16000000000000}}}{2}$$

Take the square root of \frac{1474559996800001}{16000000000000}.

$$x=\frac{-\left(-\frac{38400001}{4000000}\right)±\frac{\sqrt{1474559996800001}}{4000000}}{2}$$

The opposite of -\frac{38400001}{4000000} is \frac{38400001}{4000000}.

$$x=\frac{\frac{38400001}{4000000}±\frac{\sqrt{1474559996800001}}{4000000}}{2}$$

Now solve the equation x=\frac{\frac{38400001}{4000000}±\frac{\sqrt{1474559996800001}}{4000000}}{2} when ± is plus. Add \frac{38400001}{4000000} to \frac{\sqrt{1474559996800001}}{4000000}.

$$x=\frac{\sqrt{1474559996800001}[phone]}{2\times 4000000}$$

Divide \frac{38400001+\sqrt{1474559996800001}}{4000000} by 2.

$$x=\frac{\sqrt{1474559996800001}[phone]}{8000000}$$

Now solve the equation x=\frac{\frac{38400001}{4000000}±\frac{\sqrt{1474559996800001}}{4000000}}{2} when ± is minus. Subtract \frac{\sqrt{1474559996800001}}{4000000} from \frac{38400001}{4000000}.

$$x=\frac{38400001-\sqrt{1474559996800001}}{2\times 4000000}$$

Divide \frac{38400001-\sqrt{1474559996800001}}{4000000} by 2.

$$x=\frac{38400001-\sqrt{1474559996800001}}{8000000}$$

The equation is now solved.

$$x=\frac{\sqrt{1474559996800001}[phone]}{8000000} x=\frac{38400001-\sqrt{1474559996800001}}{8000000}$$

Steps for Completing the Square

Calculate 10 to the power of 6 and get 1000000.

$$\frac{5-x}{4\times 1000000}=9.6x-x^{2}$$

Multiply 4 and 1000000 to get 4000000.

$$\frac{5-x}{4000000}=9.6x-x^{2}$$

Divide each term of 5-x by 4000000 to get \frac{1}{800000}-\frac{1}{4000000}x.

$$\frac{1}{800000}-\frac{1}{4000000}x=9.6x-x^{2}$$

Subtract 9.6x from both sides.

$$\frac{1}{800000}-\frac{1}{4000000}x-9.6x=-x^{2}$$

Combine -\frac{1}{4000000}x and -9.6x to get -\frac{38400001}{4000000}x.

$$\frac{1}{800000}-\frac{38400001}{4000000}x=-x^{2}$$

Add x^{2} to both sides.

$$\frac{1}{800000}-\frac{38400001}{4000000}x+x^{2}=0$$

Subtract \frac{1}{800000} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{38400001}{4000000}x+x^{2}=-\frac{1}{800000}$$

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.

$$x^{2}-\frac{38400001}{4000000}x=-\frac{1}{800000}$$

Divide -\frac{38400001}{4000000}, the coefficient of the x term, by 2 to get -\frac{38400001}{8000000}. Then add the square of -\frac{38400001}{8000000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{38400001}{4000000}x+\left(-\frac{38400001}{8000000}\right)^{2}=-\frac{1}{800000}+\left(-\frac{38400001}{8000000}\right)^{2}$$

Square -\frac{38400001}{8000000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{38400001}{4000000}x+\frac{1474560076800001}{64000000000000}=-\frac{1}{800000}+\frac{1474560076800001}{64000000000000}$$

Add -\frac{1}{800000} to \frac{1474560076800001}{64000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{38400001}{4000000}x+\frac{1474560076800001}{64000000000000}=\frac{1474559996800001}{64000000000000}$$

Factor x^{2}-\frac{38400001}{4000000}x+\frac{1474560076800001}{64000000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{38400001}{8000000}\right)^{2}=\frac{1474559996800001}{64000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{38400001}{8000000}\right)^{2}}=\sqrt{\frac{1474559996800001}{64000000000000}}$$

Simplify.

$$x-\frac{38400001}{8000000}=\frac{\sqrt{1474559996800001}}{8000000} x-\frac{38400001}{8000000}=-\frac{\sqrt{1474559996800001}}{8000000}$$

Add \frac{38400001}{8000000} to both sides of the equation.

$$x=\frac{\sqrt{1474559996800001}[phone]}{8000000} x=\frac{38400001-\sqrt{1474559996800001}}{8000000}$$