Solve 2(12-2lambda+1/2)=2lambda+2lambda^2+lambda+1

$$2 ( 12 - \frac { 2 \lambda + 1 } { 2 } ) = 2 \lambda + 2 \lambda ^ { 2 } + \lambda + 1$$

Solve for λ

$$\lambda =\frac{\sqrt{201}-5}{4}\approx 2.29436172$$

$$\lambda =\frac{-\sqrt{201}-5}{4}\approx -4.79436172$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$4\left(12-\frac{2\lambda +1}{2}\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Use the distributive property to multiply 4 by 12-\frac{2\lambda +1}{2}.

$$48+4\left(-\frac{2\lambda +1}{2}\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Cancel out 2, the greatest common factor in 4 and 2.

$$48-2\left(2\lambda +1\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Use the distributive property to multiply -2 by 2\lambda +1.

$$48-4\lambda -2=4\lambda +4\lambda ^{2}+2\lambda +2$$

Subtract 2 from 48 to get 46.

$$46-4\lambda =4\lambda +4\lambda ^{2}+2\lambda +2$$

Combine 4\lambda and 2\lambda to get 6\lambda .

$$46-4\lambda =6\lambda +4\lambda ^{2}+2$$

Subtract 6\lambda from both sides.

$$46-4\lambda -6\lambda =4\lambda ^{2}+2$$

Combine -4\lambda and -6\lambda to get -10\lambda .

$$46-10\lambda =4\lambda ^{2}+2$$

Subtract 4\lambda ^{2} from both sides.

$$46-10\lambda -4\lambda ^{2}=2$$

Subtract 2 from both sides.

$$46-10\lambda -4\lambda ^{2}-2=0$$

Subtract 2 from 46 to get 44.

$$44-10\lambda -4\lambda ^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4\lambda ^{2}-10\lambda +44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -10 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-4\right)\times 44}}{2\left(-4\right)}$$

Square -10.

$$\lambda =\frac{-\left(-10\right)±\sqrt{100-4\left(-4\right)\times 44}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$\lambda =\frac{-\left(-10\right)±\sqrt{100+16\times 44}}{2\left(-4\right)}$$

Multiply 16 times 44.

$$\lambda =\frac{-\left(-10\right)±\sqrt{100+704}}{2\left(-4\right)}$$

Add 100 to 704.

$$\lambda =\frac{-\left(-10\right)±\sqrt{804}}{2\left(-4\right)}$$

Take the square root of 804.

$$\lambda =\frac{-\left(-10\right)±2\sqrt{201}}{2\left(-4\right)}$$

The opposite of -10 is 10.

$$\lambda =\frac{10±2\sqrt{201}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$\lambda =\frac{10±2\sqrt{201}}{-8}$$

Now solve the equation \lambda =\frac{10±2\sqrt{201}}{-8} when ± is plus. Add 10 to 2\sqrt{201}.

$$\lambda =\frac{2\sqrt{201}+10}{-8}$$

Divide 10+2\sqrt{201} by -8.

$$\lambda =\frac{-\sqrt{201}-5}{4}$$

Now solve the equation \lambda =\frac{10±2\sqrt{201}}{-8} when ± is minus. Subtract 2\sqrt{201} from 10.

$$\lambda =\frac{10-2\sqrt{201}}{-8}$$

Divide 10-2\sqrt{201} by -8.

$$\lambda =\frac{\sqrt{201}-5}{4}$$

The equation is now solved.

$$\lambda =\frac{-\sqrt{201}-5}{4} \lambda =\frac{\sqrt{201}-5}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$4\left(12-\frac{2\lambda +1}{2}\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Use the distributive property to multiply 4 by 12-\frac{2\lambda +1}{2}.

$$48+4\left(-\frac{2\lambda +1}{2}\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Cancel out 2, the greatest common factor in 4 and 2.

$$48-2\left(2\lambda +1\right)=4\lambda +4\lambda ^{2}+2\lambda +2$$

Use the distributive property to multiply -2 by 2\lambda +1.

$$48-4\lambda -2=4\lambda +4\lambda ^{2}+2\lambda +2$$

Subtract 2 from 48 to get 46.

$$46-4\lambda =4\lambda +4\lambda ^{2}+2\lambda +2$$

Combine 4\lambda and 2\lambda to get 6\lambda .

$$46-4\lambda =6\lambda +4\lambda ^{2}+2$$

Subtract 6\lambda from both sides.

$$46-4\lambda -6\lambda =4\lambda ^{2}+2$$

Combine -4\lambda and -6\lambda to get -10\lambda .

$$46-10\lambda =4\lambda ^{2}+2$$

Subtract 4\lambda ^{2} from both sides.

$$46-10\lambda -4\lambda ^{2}=2$$

Subtract 46 from both sides.

$$-10\lambda -4\lambda ^{2}=2-46$$

Subtract 46 from 2 to get -44.

$$-10\lambda -4\lambda ^{2}=-44$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-4\lambda ^{2}-10\lambda =-44$$

Divide both sides by -4.

$$\frac{-4\lambda ^{2}-10\lambda }{-4}=-\frac{44}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$\lambda ^{2}+\left(-\frac{10}{-4}\right)\lambda =-\frac{44}{-4}$$

Reduce the fraction \frac{-10}{-4} to lowest terms by extracting and canceling out 2.

$$\lambda ^{2}+\frac{5}{2}\lambda =-\frac{44}{-4}$$

Divide -44 by -4.

$$\lambda ^{2}+\frac{5}{2}\lambda =11$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}+\frac{5}{2}\lambda +\left(\frac{5}{4}\right)^{2}=11+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}+\frac{5}{2}\lambda +\frac{25}{16}=11+\frac{25}{16}$$

Add 11 to \frac{25}{16}.

$$\lambda ^{2}+\frac{5}{2}\lambda +\frac{25}{16}=\frac{201}{16}$$

Factor \lambda ^{2}+\frac{5}{2}\lambda +\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda +\frac{5}{4}\right)^{2}=\frac{201}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda +\frac{5}{4}\right)^{2}}=\sqrt{\frac{201}{16}}$$

Simplify.

$$\lambda +\frac{5}{4}=\frac{\sqrt{201}}{4} \lambda +\frac{5}{4}=-\frac{\sqrt{201}}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

$$\lambda =\frac{\sqrt{201}-5}{4} \lambda =\frac{-\sqrt{201}-5}{4}$$