Solve lambda^2-4999001lambda/100000+0.000225=0

$$\lambda ^ { 2 } - \frac { 4999001 \lambda } { 100000 } + 0.000225 = 0$$

Solve for λ

$$\lambda =\frac{\sqrt{24990001998001}+4999001}{200000}\approx 49.990005499$$

$$\lambda =\frac{4999001-\sqrt{24990001998001}}{200000}\approx 0.000004501$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 100000.

$$100000\lambda ^{2}-4999001\lambda +22.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100000 for a, -4999001 for b, and 22.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-4999001\right)±\sqrt{\left(-4999001\right)^{2}-4\times 100000\times 22.5}}{2\times 100000}$$

Square -4999001.

$$\lambda =\frac{-\left(-4999001\right)±\sqrt{24990010998001-4\times 100000\times 22.5}}{2\times 100000}$$

Multiply -4 times 100000.

$$\lambda =\frac{-\left(-4999001\right)±\sqrt{24990010998001-400000\times 22.5}}{2\times 100000}$$

Multiply -400000 times 22.5.

$$\lambda =\frac{-\left(-4999001\right)±\sqrt{24990010998001-9000000}}{2\times 100000}$$

Add 24990010998001 to -9000000.

$$\lambda =\frac{-\left(-4999001\right)±\sqrt{24990001998001}}{2\times 100000}$$

The opposite of -4999001 is 4999001.

$$\lambda =\frac{4999001±\sqrt{24990001998001}}{2\times 100000}$$

Multiply 2 times 100000.

$$\lambda =\frac{4999001±\sqrt{24990001998001}}{200000}$$

Now solve the equation \lambda =\frac{4999001±\sqrt{24990001998001}}{200000} when ± is plus. Add 4999001 to \sqrt{24990001998001}.

$$\lambda =\frac{\sqrt{24990001998001}+4999001}{200000}$$

Now solve the equation \lambda =\frac{4999001±\sqrt{24990001998001}}{200000} when ± is minus. Subtract \sqrt{24990001998001} from 4999001.

$$\lambda =\frac{4999001-\sqrt{24990001998001}}{200000}$$

The equation is now solved.

$$\lambda =\frac{\sqrt{24990001998001}+4999001}{200000} \lambda =\frac{4999001-\sqrt{24990001998001}}{200000}$$

Steps for Completing the Square

Multiply both sides of the equation by 100000.

$$100000\lambda ^{2}-4999001\lambda +22.5=0$$

Subtract 22.5 from both sides. Anything subtracted from zero gives its negation.

$$100000\lambda ^{2}-4999001\lambda =-22.5$$

Divide both sides by 100000.

$$\frac{100000\lambda ^{2}-4999001\lambda }{100000}=-\frac{22.5}{100000}$$

Dividing by 100000 undoes the multiplication by 100000.

$$\lambda ^{2}-\frac{4999001}{100000}\lambda =-\frac{22.5}{100000}$$

Divide -22.5 by 100000.

$$\lambda ^{2}-\frac{4999001}{100000}\lambda =-0.000225$$

Divide -\frac{4999001}{100000}, the coefficient of the x term, by 2 to get -\frac{4999001}{200000}. Then add the square of -\frac{4999001}{200000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-\frac{4999001}{100000}\lambda +\left(-\frac{4999001}{200000}\right)^{2}=-0.000225+\left(-\frac{4999001}{200000}\right)^{2}$$

Square -\frac{4999001}{200000} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-\frac{4999001}{100000}\lambda +\frac{24990010998001}{40000000000}=-0.000225+\frac{24990010998001}{40000000000}$$

Add -0.000225 to \frac{24990010998001}{40000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\lambda ^{2}-\frac{4999001}{100000}\lambda +\frac{24990010998001}{40000000000}=\frac{24990001998001}{40000000000}$$

Factor \lambda ^{2}-\frac{4999001}{100000}\lambda +\frac{24990010998001}{40000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda -\frac{4999001}{200000}\right)^{2}=\frac{24990001998001}{40000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{4999001}{200000}\right)^{2}}=\sqrt{\frac{24990001998001}{40000000000}}$$

Simplify.

$$\lambda -\frac{4999001}{200000}=\frac{\sqrt{24990001998001}}{200000} \lambda -\frac{4999001}{200000}=-\frac{\sqrt{24990001998001}}{200000}$$

Add \frac{4999001}{200000} to both sides of the equation.

$$\lambda =\frac{\sqrt{24990001998001}+4999001}{200000} \lambda =\frac{4999001-\sqrt{24990001998001}}{200000}$$