Solve x^2+2.4*10^-4x-2.88*10^-5=0

$${ x }^{ 2 } +2.4 \times { 10 }^{ -4 } x-2.88 \times { 10 }^{ -5 } = 0$$

Solve for x

$$x=\frac{3\sqrt{2001}-3}{25000}\approx 0.005247905$$

$$x=\frac{-3\sqrt{2001}-3}{25000}\approx -0.005487905$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$x^{2}+2.4\times \frac{1}{10000}x-2.88\times 10^{-5}=0$$

Multiply 2.4 and \frac{1}{10000} to get \frac{3}{12500}.

$$x^{2}+\frac{3}{12500}x-2.88\times 10^{-5}=0$$

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

$$x^{2}+\frac{3}{12500}x-2.88\times \frac{1}{100000}=0$$

Multiply 2.88 and \frac{1}{100000} to get \frac{9}{312500}.

$$x^{2}+\frac{3}{12500}x-\frac{9}{312500}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{3}{12500} for b, and -\frac{9}{312500} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{3}{12500}±\sqrt{\left(\frac{3}{12500}\right)^{2}-4\left(-\frac{9}{312500}\right)}}{2}$$

Square \frac{3}{12500} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{3}{12500}±\sqrt{\frac{9}{156250000}-4\left(-\frac{9}{312500}\right)}}{2}$$

Multiply -4 times -\frac{9}{312500}.

$$x=\frac{-\frac{3}{12500}±\sqrt{\frac{9}{156250000}+\frac{9}{78125}}}{2}$$

Add \frac{9}{156250000} to \frac{9}{78125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{3}{12500}±\sqrt{\frac{18009}{156250000}}}{2}$$

Take the square root of \frac{18009}{156250000}.

$$x=\frac{-\frac{3}{12500}±\frac{3\sqrt{2001}}{12500}}{2}$$

Now solve the equation x=\frac{-\frac{3}{12500}±\frac{3\sqrt{2001}}{12500}}{2} when ± is plus. Add -\frac{3}{12500} to \frac{3\sqrt{2001}}{12500}.

$$x=\frac{3\sqrt{2001}-3}{2\times 12500}$$

Divide \frac{-3+3\sqrt{2001}}{12500} by 2.

$$x=\frac{3\sqrt{2001}-3}{25000}$$

Now solve the equation x=\frac{-\frac{3}{12500}±\frac{3\sqrt{2001}}{12500}}{2} when ± is minus. Subtract \frac{3\sqrt{2001}}{12500} from -\frac{3}{12500}.

$$x=\frac{-3\sqrt{2001}-3}{2\times 12500}$$

Divide \frac{-3-3\sqrt{2001}}{12500} by 2.

$$x=\frac{-3\sqrt{2001}-3}{25000}$$

The equation is now solved.

$$x=\frac{3\sqrt{2001}-3}{25000} x=\frac{-3\sqrt{2001}-3}{25000}$$

Steps for Completing the Square

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$x^{2}+2.4\times \frac{1}{10000}x-2.88\times 10^{-5}=0$$

Multiply 2.4 and \frac{1}{10000} to get \frac{3}{12500}.

$$x^{2}+\frac{3}{12500}x-2.88\times 10^{-5}=0$$

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

$$x^{2}+\frac{3}{12500}x-2.88\times \frac{1}{100000}=0$$

Multiply 2.88 and \frac{1}{100000} to get \frac{9}{312500}.

$$x^{2}+\frac{3}{12500}x-\frac{9}{312500}=0$$

Add \frac{9}{312500} to both sides. Anything plus zero gives itself.

$$x^{2}+\frac{3}{12500}x=\frac{9}{312500}$$

Divide \frac{3}{12500}, the coefficient of the x term, by 2 to get \frac{3}{25000}. Then add the square of \frac{3}{25000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{12500}x+\left(\frac{3}{25000}\right)^{2}=\frac{9}{312500}+\left(\frac{3}{25000}\right)^{2}$$

Square \frac{3}{25000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{12500}x+\frac{9}{625000000}=\frac{9}{312500}+\frac{9}{625000000}$$

Add \frac{9}{312500} to \frac{9}{625000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{12500}x+\frac{9}{625000000}=\frac{18009}{625000000}$$

Factor x^{2}+\frac{3}{12500}x+\frac{9}{625000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{25000}\right)^{2}=\frac{18009}{625000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{25000}\right)^{2}}=\sqrt{\frac{18009}{625000000}}$$

Simplify.

$$x+\frac{3}{25000}=\frac{3\sqrt{2001}}{25000} x+\frac{3}{25000}=-\frac{3\sqrt{2001}}{25000}$$

Subtract \frac{3}{25000} from both sides of the equation.

$$x=\frac{3\sqrt{2001}-3}{25000} x=\frac{-3\sqrt{2001}-3}{25000}$$