Solve 2lambda^2-10lambda+18=4lambda-8

$$2 \lambda ^ { 2 } - 10 \lambda + 18 = 4 \lambda - 8$$

Solve for λ

$$\lambda =\frac{7+\sqrt{3}i}{2}\approx 3.5+0.866025404i$$

$$\lambda =\frac{-\sqrt{3}i+7}{2}\approx 3.5-0.866025404i$$

Steps Using the Quadratic Formula

Subtract 4\lambda from both sides.

$$2\lambda ^{2}-10\lambda +18-4\lambda =-8$$

Combine -10\lambda and -4\lambda to get -14\lambda .

$$2\lambda ^{2}-14\lambda +18=-8$$

Add 8 to both sides.

$$2\lambda ^{2}-14\lambda +18+8=0$$

Add 18 and 8 to get 26.

$$2\lambda ^{2}-14\lambda +26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -14 for b, and 26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 26}}{2\times 2}$$

Square -14.

$$\lambda =\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 26}}{2\times 2}$$

Multiply -4 times 2.

$$\lambda =\frac{-\left(-14\right)±\sqrt{196-8\times 26}}{2\times 2}$$

Multiply -8 times 26.

$$\lambda =\frac{-\left(-14\right)±\sqrt{196-208}}{2\times 2}$$

Add 196 to -208.

$$\lambda =\frac{-\left(-14\right)±\sqrt{-12}}{2\times 2}$$

Take the square root of -12.

$$\lambda =\frac{-\left(-14\right)±2\sqrt{3}i}{2\times 2}$$

The opposite of -14 is 14.

$$\lambda =\frac{14±2\sqrt{3}i}{2\times 2}$$

Multiply 2 times 2.

$$\lambda =\frac{14±2\sqrt{3}i}{4}$$

Now solve the equation \lambda =\frac{14±2\sqrt{3}i}{4} when ± is plus. Add 14 to 2i\sqrt{3}.

$$\lambda =\frac{14+2\sqrt{3}i}{4}$$

Divide 14+2i\sqrt{3} by 4.

$$\lambda =\frac{7+\sqrt{3}i}{2}$$

Now solve the equation \lambda =\frac{14±2\sqrt{3}i}{4} when ± is minus. Subtract 2i\sqrt{3} from 14.

$$\lambda =\frac{-2\sqrt{3}i+14}{4}$$

Divide 14-2i\sqrt{3} by 4.

$$\lambda =\frac{-\sqrt{3}i+7}{2}$$

The equation is now solved.

$$\lambda =\frac{7+\sqrt{3}i}{2} \lambda =\frac{-\sqrt{3}i+7}{2}$$

Steps for Completing the Square

Subtract 4\lambda from both sides.

$$2\lambda ^{2}-10\lambda +18-4\lambda =-8$$

Combine -10\lambda and -4\lambda to get -14\lambda .

$$2\lambda ^{2}-14\lambda +18=-8$$

Subtract 18 from both sides.

$$2\lambda ^{2}-14\lambda =-8-18$$

Subtract 18 from -8 to get -26.

$$2\lambda ^{2}-14\lambda =-26$$

Divide both sides by 2.

$$\frac{2\lambda ^{2}-14\lambda }{2}=-\frac{26}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$\lambda ^{2}+\left(-\frac{14}{2}\right)\lambda =-\frac{26}{2}$$

Divide -14 by 2.

$$\lambda ^{2}-7\lambda =-\frac{26}{2}$$

Divide -26 by 2.

$$\lambda ^{2}-7\lambda =-13$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-7\lambda +\left(-\frac{7}{2}\right)^{2}=-13+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-7\lambda +\frac{49}{4}=-13+\frac{49}{4}$$

Add -13 to \frac{49}{4}.

$$\lambda ^{2}-7\lambda +\frac{49}{4}=-\frac{3}{4}$$

Factor \lambda ^{2}-7\lambda +\frac{49}{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}.

$$\left(\lambda -\frac{7}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{7}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$\lambda -\frac{7}{2}=\frac{\sqrt{3}i}{2} \lambda -\frac{7}{2}=-\frac{\sqrt{3}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$\lambda =\frac{7+\sqrt{3}i}{2} \lambda =\frac{-\sqrt{3}i+7}{2}$$