Solve lambda^2-4999001lambda/100000+0000225=0

$$\lambda ^ { 2 } - \frac { 4999001 \lambda } { 100000 } + 0000225 = 0$$

Solve for λ

$$\lambda =\frac{4999001}{100000}=49.99001$$

$$\lambda =0$$

Steps Using Factoring

Multiply both sides of the equation by 100000.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 225=0$$

Multiply 0 and 225 to get 0.

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

Anything plus zero gives itself.

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

Factor out \lambda .

$$\lambda \left(100000\lambda -4999001\right)=0$$

To find equation solutions, solve \lambda =0 and 100000\lambda -4999001=0.

$$\lambda =0 \lambda =\frac{4999001}{100000}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 100000.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 225=0$$

Multiply 0 and 225 to get 0.

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

Anything plus zero gives itself.

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

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

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

Take the square root of \left(-4999001\right)^{2}.

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

The opposite of -4999001 is 4999001.

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

Multiply 2 times 100000.

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

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

$$\lambda =\frac{9998002}{200000}$$

Reduce the fraction \frac{9998002}{200000} to lowest terms by extracting and canceling out 2.

$$\lambda =\frac{4999001}{100000}$$

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

$$\lambda =\frac{0}{200000}$$

Divide 0 by 200000.

$$\lambda =0$$

The equation is now solved.

$$\lambda =\frac{4999001}{100000} \lambda =0$$

Steps for Completing the Square

Multiply both sides of the equation by 100000.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 0\times 225=0$$

Multiply 0 and 0 to get 0.

$$100000\lambda ^{2}-4999001\lambda +0\times 225=0$$

Multiply 0 and 225 to get 0.

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

Anything plus zero gives itself.

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

Divide both sides by 100000.

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

Dividing by 100000 undoes the multiplication by 100000.

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

Divide 0 by 100000.

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

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.

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

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

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

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}.

$$\left(\lambda -\frac{4999001}{200000}\right)^{2}=\frac{24990010998001}{40000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{4999001}{200000}\right)^{2}}=\sqrt{\frac{24990010998001}{40000000000}}$$

Simplify.

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

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

$$\lambda =\frac{4999001}{100000} \lambda =0$$