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# Factor by completing the square $\frac{x^2+x-2}{x^2+5x+6}$

## Step-by-step Solution

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###  Videos

$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$
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##  Step-by-step Solution 

Problem to solve:

$\frac{x^2+x-2}{x^2+5x+6}$

Specify the solving method

1

Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$

$\frac{x^2+x-2+\frac{1}{4}-\frac{1}{4}}{x^2+5x+6}$
2

Factor the perfect square trinomial $x^2+x+\frac{1}{4}$

$\frac{\left(x+\sqrt{\left(\frac{1}{4}\right)}\right)^2-2-\frac{1}{4}}{x^2+5x+6}$
3

Subtract the values $-2$ and $-\frac{1}{4}$

$\frac{-\frac{9}{4}+\left(x+\sqrt{\left(\frac{1}{4}\right)}\right)^2}{x^2+5x+6}$
4

Calculate the square root of $\frac{1}{4}$

$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{x^2+5x+6}$
5

Add and subtract $\displaystyle\left(\frac{b}{2a}\right)^2$

$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{x^2+5x+6+\frac{25}{4}-\frac{25}{4}}$
6

Factor the perfect square trinomial $x^2+5x+\frac{25}{4}$

$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{\left(x+\frac{5}{2}\right)^2+6-\frac{25}{4}}$
7

Subtract the values $6$ and $-\frac{25}{4}$

$\frac{-\frac{9}{4}+\left(x+\frac{1}{2}\right)^2}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$

A binomial squared (difference) is equal to the square of the first term, minus the double product of the first by the second, plus the square of the second term. In other words: $(a-b)^2=a^2-2ab+b^2$

$\frac{-\frac{9}{4}+x^2+x+\frac{1}{4}}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$

Add the values $-\frac{9}{4}$ and $\frac{1}{4}$

$\frac{-2+x^2+x}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$

A binomial squared (difference) is equal to the square of the first term, minus the double product of the first by the second, plus the square of the second term. In other words: $(a-b)^2=a^2-2ab+b^2$

$-\frac{1}{4}+x^2+5x+\frac{25}{4}$

Add the values $-\frac{1}{4}$ and $\frac{25}{4}$

$6+x^2+5x$
8

Expand

$\frac{-2+x^2+x}{6+x^2+5x}$
9

Divide $-2+x^2+x$ by $6+x^2+5x$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}{\phantom{;}1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+6\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}\underline{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-x^{2}-5x\phantom{;}-6\phantom{;}\phantom{;};}-4x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$
10

Resulting polynomial

$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$

$1+\frac{-4x-8}{-\frac{1}{4}+\left(x+\frac{5}{2}\right)^2}$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SimplifyFactorFind the integralFind the derivativeFind (x^2+x)/(x^2+5x) using the definitionFind the rootsFind break even pointsFind the discriminant

SnapXam A2

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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Main topic:

Polynomial Factorization

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