Solve 1.210^3x^2-15.4x+3.8410^-2=0

$$1.2 { 10 }^{ 3 } { x }^{ 2 } -15.4x+3.84 { 10 }^{ -2 } = 0$$

Solve for x

$$x=\frac{\sqrt{1321}+77}{12000}\approx 0.009445464$$

$$x=\frac{77-\sqrt{1321}}{12000}\approx 0.00338787$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 3 and get 1000.

$$1.2\times 1000x^{2}-15.4x+3.84\times 10^{-2}=0$$

Multiply 1.2 and 1000 to get 1200.

$$1200x^{2}-15.4x+3.84\times 10^{-2}=0$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$1200x^{2}-15.4x+3.84\times \frac{1}{100}=0$$

Multiply 3.84 and \frac{1}{100} to get \frac{24}{625}.

$$1200x^{2}-15.4x+\frac{24}{625}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1200 for a, -15.4 for b, and \frac{24}{625} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15.4\right)±\sqrt{\left(-15.4\right)^{2}-4\times 1200\times \frac{24}{625}}}{2\times 1200}$$

Square -15.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-15.4\right)±\sqrt{237.16-4\times 1200\times \frac{24}{625}}}{2\times 1200}$$

Multiply -4 times 1200.

$$x=\frac{-\left(-15.4\right)±\sqrt{237.16-4800\times \frac{24}{625}}}{2\times 1200}$$

Multiply -4800 times \frac{24}{625}.

$$x=\frac{-\left(-15.4\right)±\sqrt{\frac{5929-4608}{25}}}{2\times 1200}$$

Add 237.16 to -\frac{4608}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-15.4\right)±\sqrt{\frac{1321}{25}}}{2\times 1200}$$

Take the square root of \frac{1321}{25}.

$$x=\frac{-\left(-15.4\right)±\frac{\sqrt{1321}}{5}}{2\times 1200}$$

The opposite of -15.4 is 15.4.

$$x=\frac{15.4±\frac{\sqrt{1321}}{5}}{2\times 1200}$$

Multiply 2 times 1200.

$$x=\frac{15.4±\frac{\sqrt{1321}}{5}}{2400}$$

Now solve the equation x=\frac{15.4±\frac{\sqrt{1321}}{5}}{2400} when ± is plus. Add 15.4 to \frac{\sqrt{1321}}{5}.

$$x=\frac{\sqrt{1321}+77}{5\times 2400}$$

Divide \frac{77+\sqrt{1321}}{5} by 2400.

$$x=\frac{\sqrt{1321}+77}{12000}$$

Now solve the equation x=\frac{15.4±\frac{\sqrt{1321}}{5}}{2400} when ± is minus. Subtract \frac{\sqrt{1321}}{5} from 15.4.

$$x=\frac{77-\sqrt{1321}}{5\times 2400}$$

Divide \frac{77-\sqrt{1321}}{5} by 2400.

$$x=\frac{77-\sqrt{1321}}{12000}$$

The equation is now solved.

$$x=\frac{\sqrt{1321}+77}{12000} x=\frac{77-\sqrt{1321}}{12000}$$

Steps for Completing the Square

Calculate 10 to the power of 3 and get 1000.

$$1.2\times 1000x^{2}-15.4x+3.84\times 10^{-2}=0$$

Multiply 1.2 and 1000 to get 1200.

$$1200x^{2}-15.4x+3.84\times 10^{-2}=0$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$1200x^{2}-15.4x+3.84\times \frac{1}{100}=0$$

Multiply 3.84 and \frac{1}{100} to get \frac{24}{625}.

$$1200x^{2}-15.4x+\frac{24}{625}=0$$

Subtract \frac{24}{625} from both sides. Anything subtracted from zero gives its negation.

$$1200x^{2}-15.4x=-\frac{24}{625}$$

Divide both sides by 1200.

$$\frac{1200x^{2}-15.4x}{1200}=-\frac{\frac{24}{625}}{1200}$$

Dividing by 1200 undoes the multiplication by 1200.

$$x^{2}+\left(-\frac{15.4}{1200}\right)x=-\frac{\frac{24}{625}}{1200}$$

Divide -15.4 by 1200.

$$x^{2}-\frac{77}{6000}x=-\frac{\frac{24}{625}}{1200}$$

Divide -\frac{24}{625} by 1200.

$$x^{2}-\frac{77}{6000}x=-\frac{1}{31250}$$

Divide -\frac{77}{6000}, the coefficient of the x term, by 2 to get -\frac{77}{12000}. Then add the square of -\frac{77}{12000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{77}{6000}x+\left(-\frac{77}{12000}\right)^{2}=-\frac{1}{31250}+\left(-\frac{77}{12000}\right)^{2}$$

Square -\frac{77}{12000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{77}{6000}x+\frac{5929}{144000000}=-\frac{1}{31250}+\frac{5929}{144000000}$$

Add -\frac{1}{31250} to \frac{5929}{144000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{77}{6000}x+\frac{5929}{144000000}=\frac{1321}{144000000}$$

Factor x^{2}-\frac{77}{6000}x+\frac{5929}{144000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{77}{12000}\right)^{2}=\frac{1321}{144000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{77}{12000}\right)^{2}}=\sqrt{\frac{1321}{144000000}}$$

Simplify.

$$x-\frac{77}{12000}=\frac{\sqrt{1321}}{12000} x-\frac{77}{12000}=-\frac{\sqrt{1321}}{12000}$$

Add \frac{77}{12000} to both sides of the equation.

$$x=\frac{\sqrt{1321}+77}{12000} x=\frac{77-\sqrt{1321}}{12000}$$