Solve 4.4x^2+0.0305x-1.8596*10^-4=0

$$4.4 { x }^{ 2 } +0.0305x-1.8596 \times { 10 }^{ -4 } = 0$$

Solve for x

$$x=\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}\approx 0.003901327$$

$$x=-\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}\approx -0.010833145$$

Steps Using the Quadratic Formula

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

$$4.4x^{2}+0.0305x-1.8596\times \frac{1}{10000}=0$$

Multiply 1.8596 and \frac{1}{10000} to get \frac{4649}{25000000}.

$$4.4x^{2}+0.0305x-\frac{4649}{25000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.4 for a, 0.0305 for b, and -\frac{4649}{25000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.0305±\sqrt{0.0305^{2}-4\times 4.4\left(-\frac{4649}{25000000}\right)}}{2\times 4.4}$$

Square 0.0305 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.0305±\sqrt{0.00093025-4\times 4.4\left(-\frac{4649}{25000000}\right)}}{2\times 4.4}$$

Multiply -4 times 4.4.

$$x=\frac{-0.0305±\sqrt{0.00093025-17.6\left(-\frac{4649}{25000000}\right)}}{2\times 4.4}$$

Multiply -17.6 times -\frac{4649}{25000000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0305±\sqrt{0.00093025+\frac{51139}{15625000}}}{2\times 4.4}$$

Add 0.00093025 to \frac{51139}{15625000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0305±\sqrt{\frac{2101573}{500000000}}}{2\times 4.4}$$

Take the square root of \frac{2101573}{500000000}.

$$x=\frac{-0.0305±\frac{\sqrt{10507865}}{50000}}{2\times 4.4}$$

Multiply 2 times 4.4.

$$x=\frac{-0.0305±\frac{\sqrt{10507865}}{50000}}{8.8}$$

Now solve the equation x=\frac{-0.0305±\frac{\sqrt{10507865}}{50000}}{8.8} when ± is plus. Add -0.0305 to \frac{\sqrt{10507865}}{50000}.

$$x=\frac{\frac{\sqrt{10507865}}{50000}-\frac{61}{2000}}{8.8}$$

Divide -\frac{61}{2000}+\frac{\sqrt{10507865}}{50000} by 8.8 by multiplying -\frac{61}{2000}+\frac{\sqrt{10507865}}{50000} by the reciprocal of 8.8.

$$x=\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}$$

Now solve the equation x=\frac{-0.0305±\frac{\sqrt{10507865}}{50000}}{8.8} when ± is minus. Subtract \frac{\sqrt{10507865}}{50000} from -0.0305.

$$x=\frac{-\frac{\sqrt{10507865}}{50000}-\frac{61}{2000}}{8.8}$$

Divide -\frac{61}{2000}-\frac{\sqrt{10507865}}{50000} by 8.8 by multiplying -\frac{61}{2000}-\frac{\sqrt{10507865}}{50000} by the reciprocal of 8.8.

$$x=-\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}$$

The equation is now solved.

$$x=\frac{\sqrt{10507865}}{440000}-\frac{61}{17600} x=-\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}$$

Steps for Completing the Square

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

$$4.4x^{2}+0.0305x-1.8596\times \frac{1}{10000}=0$$

Multiply 1.8596 and \frac{1}{10000} to get \frac{4649}{25000000}.

$$4.4x^{2}+0.0305x-\frac{4649}{25000000}=0$$

Add \frac{4649}{25000000} to both sides. Anything plus zero gives itself.

$$4.4x^{2}+0.0305x=\frac{4649}{25000000}$$

Divide both sides of the equation by 4.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.4x^{2}+0.0305x}{4.4}=\frac{\frac{4649}{25000000}}{4.4}$$

Dividing by 4.4 undoes the multiplication by 4.4.

$$x^{2}+\frac{0.0305}{4.4}x=\frac{\frac{4649}{25000000}}{4.4}$$

Divide 0.0305 by 4.4 by multiplying 0.0305 by the reciprocal of 4.4.

$$x^{2}+\frac{61}{8800}x=\frac{\frac{4649}{25000000}}{4.4}$$

Divide \frac{4649}{25000000} by 4.4 by multiplying \frac{4649}{25000000} by the reciprocal of 4.4.

$$x^{2}+\frac{61}{8800}x=\frac{4649}{110000000}$$

Divide \frac{61}{8800}, the coefficient of the x term, by 2 to get \frac{61}{17600}. Then add the square of \frac{61}{17600} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{61}{8800}x+\frac{61}{17600}^{2}=\frac{4649}{110000000}+\frac{61}{17600}^{2}$$

Square \frac{61}{17600} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{61}{8800}x+\frac{3721}{309760000}=\frac{4649}{110000000}+\frac{3721}{309760000}$$

Add \frac{4649}{110000000} to \frac{3721}{309760000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{61}{8800}x+\frac{3721}{309760000}=\frac{2101573}{38720000000}$$

Factor x^{2}+\frac{61}{8800}x+\frac{3721}{309760000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{61}{17600}\right)^{2}=\frac{2101573}{38720000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{61}{17600}\right)^{2}}=\sqrt{\frac{2101573}{38720000000}}$$

Simplify.

$$x+\frac{61}{17600}=\frac{\sqrt{10507865}}{440000} x+\frac{61}{17600}=-\frac{\sqrt{10507865}}{440000}$$

Subtract \frac{61}{17600} from both sides of the equation.

$$x=\frac{\sqrt{10507865}}{440000}-\frac{61}{17600} x=-\frac{\sqrt{10507865}}{440000}-\frac{61}{17600}$$