Solve left(lambda^2-2right)1+left(lambda-5right)(-2)=2left(lambda+2right)

$$\left( { \lambda }^{ 2 } -2 \right) 1+ \left( \lambda -5 \right) (-2) = 2 \left( \lambda +2 \right)$$

Solve for λ

$$\lambda =2$$

Steps Using Factoring

Use the distributive property to multiply \lambda ^{2}-2 by 1.

$$\lambda ^{2}-2+\left(\lambda -5\right)\left(-2\right)=2\left(\lambda +2\right)$$

Use the distributive property to multiply \lambda -5 by -2.

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

Add -2 and 10 to get 8.

$$\lambda ^{2}+8-2\lambda =2\left(\lambda +2\right)$$

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

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

Subtract 2\lambda from both sides.

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

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

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

Subtract 4 from both sides.

$$\lambda ^{2}+8-4\lambda -4=0$$

Subtract 4 from 8 to get 4.

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

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

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

To solve the equation, factor \lambda ^{2}-4\lambda +4 using formula \lambda ^{2}+\left(a+b\right)\lambda +ab=\left(\lambda +a\right)\left(\lambda +b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

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

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -4.

$$a=-2 b=-2$$

Rewrite factored expression \left(\lambda +a\right)\left(\lambda +b\right) using the obtained values.

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

Rewrite as a binomial square.

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

To find equation solution, solve \lambda -2=0.

$$\lambda =2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply \lambda ^{2}-2 by 1.

$$\lambda ^{2}-2+\left(\lambda -5\right)\left(-2\right)=2\left(\lambda +2\right)$$

Use the distributive property to multiply \lambda -5 by -2.

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

Add -2 and 10 to get 8.

$$\lambda ^{2}+8-2\lambda =2\left(\lambda +2\right)$$

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

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

Subtract 2\lambda from both sides.

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

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

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

Subtract 4 from both sides.

$$\lambda ^{2}+8-4\lambda -4=0$$

Subtract 4 from 8 to get 4.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as \lambda ^{2}+a\lambda +b\lambda +4. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 4=4$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 4.

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

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -4.

$$a=-2 b=-2$$

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

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

Factor out \lambda in the first and -2 in the second group.

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

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

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

Rewrite as a binomial square.

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

To find equation solution, solve \lambda -2=0.

$$\lambda =2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \lambda ^{2}-2 by 1.

$$\lambda ^{2}-2+\left(\lambda -5\right)\left(-2\right)=2\left(\lambda +2\right)$$

Use the distributive property to multiply \lambda -5 by -2.

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

Add -2 and 10 to get 8.

$$\lambda ^{2}+8-2\lambda =2\left(\lambda +2\right)$$

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

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

Subtract 2\lambda from both sides.

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

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

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

Subtract 4 from both sides.

$$\lambda ^{2}+8-4\lambda -4=0$$

Subtract 4 from 8 to get 4.

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

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.

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

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

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

Square -4.

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

Multiply -4 times 4.

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

Add 16 to -16.

$$\lambda =\frac{-\left(-4\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$\lambda =-\frac{-4}{2}$$

The opposite of -4 is 4.

$$\lambda =\frac{4}{2}$$

Divide 4 by 2.

$$\lambda =2$$

Steps for Completing the Square

Use the distributive property to multiply \lambda ^{2}-2 by 1.

$$\lambda ^{2}-2+\left(\lambda -5\right)\left(-2\right)=2\left(\lambda +2\right)$$

Use the distributive property to multiply \lambda -5 by -2.

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

Add -2 and 10 to get 8.

$$\lambda ^{2}+8-2\lambda =2\left(\lambda +2\right)$$

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

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

Subtract 2\lambda from both sides.

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

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

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

Subtract 8 from both sides.

$$\lambda ^{2}-4\lambda =4-8$$

Subtract 8 from 4 to get -4.

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

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

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

Square -2.

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

Add -4 to 4.

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

Factor \lambda ^{2}-4\lambda +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}.

$$\left(\lambda -2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -2\right)^{2}}=\sqrt{0}$$

Simplify.

$$\lambda -2=0 \lambda -2=0$$

Add 2 to both sides of the equation.

$$\lambda =2 \lambda =2$$

The equation is now solved. Solutions are the same.

$$\lambda =2$$