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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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Final Answer

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

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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 $-\frac{25}{4}$

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

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

Final Answer

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

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 definitionSolve by quadratic formula (general formula)Find the rootsFind break even pointsFind the discriminant

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Function Plot

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

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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

How to improve your answer:

Main Topic: Polynomial Factorization

They are a group of techniques that help us rewrite polynomial expressions as a product of factors.

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