Solve left(3lambda-6right)left(3+6lambdaright)+left(2lambda+1right)left(2-lambdaright)=0

$$\left( 3 \lambda -6 \right) \left( 3+6 \lambda \right) + \left( 2 \lambda +1 \right) \left( 2- \lambda \right) = 0$$

Solve for λ

$$\lambda =-\frac{1}{2}=-0.5$$

$$\lambda =2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 3\lambda -6 by 3+6\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+\left(2\lambda +1\right)\left(2-\lambda \right)=0$$

Use the distributive property to multiply 2\lambda +1 by 2-\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+3\lambda -2\lambda ^{2}+2=0$$

Combine -27\lambda and 3\lambda to get -24\lambda .

$$-24\lambda +18\lambda ^{2}-18-2\lambda ^{2}+2=0$$

Combine 18\lambda ^{2} and -2\lambda ^{2} to get 16\lambda ^{2}.

$$-24\lambda +16\lambda ^{2}-18+2=0$$

Add -18 and 2 to get -16.

$$-24\lambda +16\lambda ^{2}-16=0$$

Divide both sides by 8.

$$-3\lambda +2\lambda ^{2}-2=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$2\lambda ^{2}-3\lambda -2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2\lambda ^{2}+a\lambda +b\lambda -2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-2\right)=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

Rewrite 2\lambda ^{2}-3\lambda -2 as \left(2\lambda ^{2}-4\lambda \right)+\left(\lambda -2\right).

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

Factor out 2\lambda in 2\lambda ^{2}-4\lambda .

$$2\lambda \left(\lambda -2\right)+\lambda -2$$

Factor out common term \lambda -2 by using distributive property.

$$\left(\lambda -2\right)\left(2\lambda +1\right)$$

To find equation solutions, solve \lambda -2=0 and 2\lambda +1=0.

$$\lambda =2 \lambda =-\frac{1}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3\lambda -6 by 3+6\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+\left(2\lambda +1\right)\left(2-\lambda \right)=0$$

Use the distributive property to multiply 2\lambda +1 by 2-\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+3\lambda -2\lambda ^{2}+2=0$$

Combine -27\lambda and 3\lambda to get -24\lambda .

$$-24\lambda +18\lambda ^{2}-18-2\lambda ^{2}+2=0$$

Combine 18\lambda ^{2} and -2\lambda ^{2} to get 16\lambda ^{2}.

$$-24\lambda +16\lambda ^{2}-18+2=0$$

Add -18 and 2 to get -16.

$$-24\lambda +16\lambda ^{2}-16=0$$

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.

$$16\lambda ^{2}-24\lambda -16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -24 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\left(-16\right)}}{2\times 16}$$

Square -24.

$$\lambda =\frac{-\left(-24\right)±\sqrt{576-4\times 16\left(-16\right)}}{2\times 16}$$

Multiply -4 times 16.

$$\lambda =\frac{-\left(-24\right)±\sqrt{576-64\left(-16\right)}}{2\times 16}$$

Multiply -64 times -16.

$$\lambda =\frac{-\left(-24\right)±\sqrt{576+1024}}{2\times 16}$$

Add 576 to 1024.

$$\lambda =\frac{-\left(-24\right)±\sqrt{1600}}{2\times 16}$$

Take the square root of 1600.

$$\lambda =\frac{-\left(-24\right)±40}{2\times 16}$$

The opposite of -24 is 24.

$$\lambda =\frac{24±40}{2\times 16}$$

Multiply 2 times 16.

$$\lambda =\frac{24±40}{32}$$

Now solve the equation \lambda =\frac{24±40}{32} when ± is plus. Add 24 to 40.

$$\lambda =\frac{64}{32}$$

Divide 64 by 32.

$$\lambda =2$$

Now solve the equation \lambda =\frac{24±40}{32} when ± is minus. Subtract 40 from 24.

$$\lambda =-\frac{16}{32}$$

Reduce the fraction \frac{-16}{32} to lowest terms by extracting and canceling out 16.

$$\lambda =-\frac{1}{2}$$

The equation is now solved.

$$\lambda =2 \lambda =-\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3\lambda -6 by 3+6\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+\left(2\lambda +1\right)\left(2-\lambda \right)=0$$

Use the distributive property to multiply 2\lambda +1 by 2-\lambda and combine like terms.

$$-27\lambda +18\lambda ^{2}-18+3\lambda -2\lambda ^{2}+2=0$$

Combine -27\lambda and 3\lambda to get -24\lambda .

$$-24\lambda +18\lambda ^{2}-18-2\lambda ^{2}+2=0$$

Combine 18\lambda ^{2} and -2\lambda ^{2} to get 16\lambda ^{2}.

$$-24\lambda +16\lambda ^{2}-18+2=0$$

Add -18 and 2 to get -16.

$$-24\lambda +16\lambda ^{2}-16=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$-24\lambda +16\lambda ^{2}=16$$

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.

$$16\lambda ^{2}-24\lambda =16$$

Divide both sides by 16.

$$\frac{16\lambda ^{2}-24\lambda }{16}=\frac{16}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$\lambda ^{2}+\left(-\frac{24}{16}\right)\lambda =\frac{16}{16}$$

Reduce the fraction \frac{-24}{16} to lowest terms by extracting and canceling out 8.

$$\lambda ^{2}-\frac{3}{2}\lambda =\frac{16}{16}$$

Divide 16 by 16.

$$\lambda ^{2}-\frac{3}{2}\lambda =1$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-\frac{3}{2}\lambda +\left(-\frac{3}{4}\right)^{2}=1+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-\frac{3}{2}\lambda +\frac{9}{16}=1+\frac{9}{16}$$

Add 1 to \frac{9}{16}.

$$\lambda ^{2}-\frac{3}{2}\lambda +\frac{9}{16}=\frac{25}{16}$$

Factor \lambda ^{2}-\frac{3}{2}\lambda +\frac{9}{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}.

$$\left(\lambda -\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$\lambda -\frac{3}{4}=\frac{5}{4} \lambda -\frac{3}{4}=-\frac{5}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$\lambda =2 \lambda =-\frac{1}{2}$$