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Find the roots of $\frac{x^2+x-2}{x^2+5x+6}$

Step-by-step Solution

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Final Answer

$x=1$
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Step-by-step Solution

Specify the solving method

1

Find the roots of the equation using the Quadratic Formula

$\frac{x^2+x-2}{x^2+5x+6}=0$
2

Factor the trinomial $x^2+5x+6$ finding two numbers that multiply to form $6$ and added form $5$

$\begin{matrix}\left(2\right)\left(3\right)=6\\ \left(2\right)+\left(3\right)=5\end{matrix}$
3

Thus

$\frac{x^2+x-2}{\left(x+2\right)\left(x+3\right)}=0$
4

Factor the trinomial $x^2+x-2$ finding two numbers that multiply to form $-2$ and added form $1$

$\begin{matrix}\left(-1\right)\left(2\right)=-2\\ \left(-1\right)+\left(2\right)=1\end{matrix}$
5

Thus

$\frac{\left(x-1\right)\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}=0$
6

Simplifying

$\frac{x-1}{x+3}=0$
7

Multiply both sides of the equation by $x+3$

$x-1=0$
8

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-1$ from both sides of the equation

$x-1+1=0+1$
9

Canceling terms on both sides

$x=1$

Verify that the solutions obtained are valid in the initial equation

10

The valid solutions to the equation are the ones that, when replaced in the original equation, don't make any denominator equal to $0$, since division by zero is not allowed

$x=1$

Final Answer

$x=1$

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

Solve for xFind the rootsSolve by factoringSolve by completing the squareFind break even pointsFind the discriminant

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Function Plot

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

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1
2
3
4
5
6
7
8
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: Equations

In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions.

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