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x^2+2x-5<0
Solving inequalities/ x^2+2x-5<0

x^2+2x-5<0 inequation

A inequation with variable

The solution

You have entered [src] 
 2              
x  + 2*x - 5 < 0
x2+2x−5<0x^{2} + 2 x - 5 < 0 
x^2 + 2*x - 1*5 < 0
Detail solution
Given the inequality:
x2+2x−5<0x^{2} + 2 x - 5 < 0
To solve this inequality, we must first solve the corresponding equation:
x2+2x−5=0x^{2} + 2 x - 5 = 0
Solve:
This equation is of the form
a x2+b x+c=0a\ x^2 + b\ x + c = 0
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=1a = 1
b=2b = 2
c=−5c = -5
, then
D=b2−4 a c=D = b^2 - 4\ a\ c =
22−1⋅4(−5)=242^{2} - 1 \cdot 4 \left(-5\right) = 24
Because D > 0, then the equation has two roots.
x1=(−b+D)2ax_1 = \frac{(-b + \sqrt{D})}{2 a}
x2=(−b−D)2ax_2 = \frac{(-b - \sqrt{D})}{2 a}
or
x1=−1+6x_{1} = -1 + \sqrt{6}
Simplify
x2=−6−1x_{2} = - \sqrt{6} - 1
Simplify
x1=−1+6x_{1} = -1 + \sqrt{6}
x2=−6−1x_{2} = - \sqrt{6} - 1
x1=−1+6x_{1} = -1 + \sqrt{6}
x2=−6−1x_{2} = - \sqrt{6} - 1
This roots
x2=−6−1x_{2} = - \sqrt{6} - 1
x1=−1+6x_{1} = -1 + \sqrt{6}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x2x_{0} < x_{2}
For example, let's take the point
x0=x2−110x_{0} = x_{2} - \frac{1}{10}
=
(−6−1)−110\left(- \sqrt{6} - 1\right) - \frac{1}{10}
=
−6−1110- \sqrt{6} - \frac{11}{10}
substitute to the expression
x2+2x−5<0x^{2} + 2 x - 5 < 0
2(−6−1110)−5+(−6−1110)2<02 \left(- \sqrt{6} - \frac{11}{10}\right) - 5 + \left(- \sqrt{6} - \frac{11}{10}\right)^{2} < 0
                     2              
  36   /  11     ___\        ___    
- -- + |- -- - \/ 6 |  - 2*\/ 6  < 0
  5    \  10        /

but
                     2              
  36   /  11     ___\        ___    
- -- + |- -- - \/ 6 |  - 2*\/ 6  > 0
  5    \  10        /

Then
x<−6−1x < - \sqrt{6} - 1
no execute
one of the solutions of our inequality is:
x>−6−1∧x<−1+6x > - \sqrt{6} - 1 \wedge x < -1 + \sqrt{6}
         _____  
        /     \  
-------ο-------ο-------
       x_2      x_1
Solving inequality on a graph
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-2525
Rapid solution [src] 
   /           ___         ___    \
And\x < -1 + \/ 6 , -1 - \/ 6  < x/
x<−1+6∧−6−1<xx < -1 + \sqrt{6} \wedge - \sqrt{6} - 1 < x 
(x < -1 + sqrt(6))∧(-1 - sqrt(6) < x)
Rapid solution 2 [src] 
        ___         ___ 
(-1 - \/ 6 , -1 + \/ 6 )
x in (−6−1,−1+6)x\ in\ \left(- \sqrt{6} - 1, -1 + \sqrt{6}\right) 
x in Interval.open(-sqrt(6) - 1, -1 + sqrt(6))
The graph
x^2+2x-5<0 inequation
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