Solve 3.4x^2+5.8*10^-2x+2.4*10^-4=0

$$3.4 { x }^{ 2 } +5.8 \times { 10 }^{ -2 } x+2.4 \times { 10 }^{ -4 } =0$$

Solve for x

$$x=-\frac{3}{425}\approx -0.007058824$$

$$x=-0.01$$

Steps Using the Quadratic Formula

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

$$3.4x^{2}+5.8\times \frac{1}{100}x+2.4\times 10^{-4}=0$$

Multiply 5.8 and \frac{1}{100} to get \frac{29}{500}.

$$3.4x^{2}+\frac{29}{500}x+2.4\times 10^{-4}=0$$

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

$$3.4x^{2}+\frac{29}{500}x+2.4\times \frac{1}{10000}=0$$

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

$$3.4x^{2}+\frac{29}{500}x+\frac{3}{12500}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.4 for a, \frac{29}{500} for b, and \frac{3}{12500} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{29}{500}±\sqrt{\left(\frac{29}{500}\right)^{2}-4\times 3.4\times \frac{3}{12500}}}{2\times 3.4}$$

Square \frac{29}{500} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{29}{500}±\sqrt{\frac{841}{250000}-4\times 3.4\times \frac{3}{12500}}}{2\times 3.4}$$

Multiply -4 times 3.4.

$$x=\frac{-\frac{29}{500}±\sqrt{\frac{841}{250000}-13.6\times \frac{3}{12500}}}{2\times 3.4}$$

Multiply -13.6 times \frac{3}{12500} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{29}{500}±\sqrt{\frac{841}{250000}-\frac{51}{15625}}}{2\times 3.4}$$

Add \frac{841}{250000} to -\frac{51}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{29}{500}±\sqrt{\frac{1}{10000}}}{2\times 3.4}$$

Take the square root of \frac{1}{10000}.

$$x=\frac{-\frac{29}{500}±\frac{1}{100}}{2\times 3.4}$$

Multiply 2 times 3.4.

$$x=\frac{-\frac{29}{500}±\frac{1}{100}}{6.8}$$

Now solve the equation x=\frac{-\frac{29}{500}±\frac{1}{100}}{6.8} when ± is plus. Add -\frac{29}{500} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{6}{125}}{6.8}$$

Divide -\frac{6}{125} by 6.8 by multiplying -\frac{6}{125} by the reciprocal of 6.8.

$$x=-\frac{3}{425}$$

Now solve the equation x=\frac{-\frac{29}{500}±\frac{1}{100}}{6.8} when ± is minus. Subtract \frac{1}{100} from -\frac{29}{500} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{17}{250}}{6.8}$$

Divide -\frac{17}{250} by 6.8 by multiplying -\frac{17}{250} by the reciprocal of 6.8.

$$x=-\frac{1}{100}$$

The equation is now solved.

$$x=-\frac{3}{425} x=-\frac{1}{100}$$

Steps for Completing the Square

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

$$3.4x^{2}+5.8\times \frac{1}{100}x+2.4\times 10^{-4}=0$$

Multiply 5.8 and \frac{1}{100} to get \frac{29}{500}.

$$3.4x^{2}+\frac{29}{500}x+2.4\times 10^{-4}=0$$

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

$$3.4x^{2}+\frac{29}{500}x+2.4\times \frac{1}{10000}=0$$

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

$$3.4x^{2}+\frac{29}{500}x+\frac{3}{12500}=0$$

Subtract \frac{3}{12500} from both sides. Anything subtracted from zero gives its negation.

$$3.4x^{2}+\frac{29}{500}x=-\frac{3}{12500}$$

Divide both sides of the equation by 3.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.4x^{2}+\frac{29}{500}x}{3.4}=-\frac{\frac{3}{12500}}{3.4}$$

Dividing by 3.4 undoes the multiplication by 3.4.

$$x^{2}+\frac{\frac{29}{500}}{3.4}x=-\frac{\frac{3}{12500}}{3.4}$$

Divide \frac{29}{500} by 3.4 by multiplying \frac{29}{500} by the reciprocal of 3.4.

$$x^{2}+\frac{29}{1700}x=-\frac{\frac{3}{12500}}{3.4}$$

Divide -\frac{3}{12500} by 3.4 by multiplying -\frac{3}{12500} by the reciprocal of 3.4.

$$x^{2}+\frac{29}{1700}x=-\frac{3}{42500}$$

Divide \frac{29}{1700}, the coefficient of the x term, by 2 to get \frac{29}{3400}. Then add the square of \frac{29}{3400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{29}{1700}x+\left(\frac{29}{3400}\right)^{2}=-\frac{3}{42500}+\left(\frac{29}{3400}\right)^{2}$$

Square \frac{29}{3400} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{29}{1700}x+\frac{841}{11560000}=-\frac{3}{42500}+\frac{841}{11560000}$$

Add -\frac{3}{42500} to \frac{841}{11560000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{29}{1700}x+\frac{841}{11560000}=\frac{1}{462400}$$

Factor x^{2}+\frac{29}{1700}x+\frac{841}{11560000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{29}{3400}\right)^{2}=\frac{1}{462400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{29}{3400}\right)^{2}}=\sqrt{\frac{1}{462400}}$$

Simplify.

$$x+\frac{29}{3400}=\frac{1}{680} x+\frac{29}{3400}=-\frac{1}{680}$$

Subtract \frac{29}{3400} from both sides of the equation.

$$x=-\frac{3}{425} x=-\frac{1}{100}$$