Solve left(lambda-8right)left(lambda-17right)=100

$$\left( \lambda -8 \right) \left( \lambda -17 \right) = 100$$

Solve for λ

$$\lambda =\frac{\sqrt{481}+25}{2}\approx 23.4658561$$

$$\lambda =\frac{25-\sqrt{481}}{2}\approx 1.5341439$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \lambda -8 by \lambda -17 and combine like terms.

$$\lambda ^{2}-25\lambda +136=100$$

Subtract 100 from both sides.

$$\lambda ^{2}-25\lambda +136-100=0$$

Subtract 100 from 136 to get 36.

$$\lambda ^{2}-25\lambda +36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -25 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 36}}{2}$$

Square -25.

$$\lambda =\frac{-\left(-25\right)±\sqrt{625-4\times 36}}{2}$$

Multiply -4 times 36.

$$\lambda =\frac{-\left(-25\right)±\sqrt{625-144}}{2}$$

Add 625 to -144.

$$\lambda =\frac{-\left(-25\right)±\sqrt{481}}{2}$$

The opposite of -25 is 25.

$$\lambda =\frac{25±\sqrt{481}}{2}$$

Now solve the equation \lambda =\frac{25±\sqrt{481}}{2} when ± is plus. Add 25 to \sqrt{481}.

$$\lambda =\frac{\sqrt{481}+25}{2}$$

Now solve the equation \lambda =\frac{25±\sqrt{481}}{2} when ± is minus. Subtract \sqrt{481} from 25.

$$\lambda =\frac{25-\sqrt{481}}{2}$$

The equation is now solved.

$$\lambda =\frac{\sqrt{481}+25}{2} \lambda =\frac{25-\sqrt{481}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply \lambda -8 by \lambda -17 and combine like terms.

$$\lambda ^{2}-25\lambda +136=100$$

Subtract 136 from both sides.

$$\lambda ^{2}-25\lambda =100-136$$

Subtract 136 from 100 to get -36.

$$\lambda ^{2}-25\lambda =-36$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-25\lambda +\left(-\frac{25}{2}\right)^{2}=-36+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-25\lambda +\frac{625}{4}=-36+\frac{625}{4}$$

Add -36 to \frac{625}{4}.

$$\lambda ^{2}-25\lambda +\frac{625}{4}=\frac{481}{4}$$

Factor \lambda ^{2}-25\lambda +\frac{625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda -\frac{25}{2}\right)^{2}=\frac{481}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{25}{2}\right)^{2}}=\sqrt{\frac{481}{4}}$$

Simplify.

$$\lambda -\frac{25}{2}=\frac{\sqrt{481}}{2} \lambda -\frac{25}{2}=-\frac{\sqrt{481}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$\lambda =\frac{\sqrt{481}+25}{2} \lambda =\frac{25-\sqrt{481}}{2}$$