Quadratic inequalities
      Solving quadratic inequalities
      Graphic solution of quadratic inequalities
Quadratic inequalities
To solve a quadratic inequality we can examine the sign of the equivalent quadratic function.
The x-intercepts or roots are the points where a quadratic function changes the sign. The x-intercepts determine the three intervals on the x-axis in which the function is above or under the x-axis, that is, where the function is positive or negative.
Graphic solution of quadratic inequalities
Example:  Solve the inequality   - x2 + 2x + 3 0.
Solution:  Solve the quadratic equation  ax2 + bx + c = 0 to get the boundary points.
The zeroes or roots of equivalent function (see the graph below) are the endpoints of the intervals and are included in the solution.
The turning point V (x0, y0),
 
The roots,  - x2 + 2x + 3 = 0
Solution:  
Example:  Solve the inequality    
Solution:  A fraction is negative if the numerator and the denominator have opposite signs.
Thus, we have to solve the two systems of inequalities
a)   x  - 2 < 0      and      b)    x - 2 > 0
      2x + 3 > 0                     2x + 3 < 0
                                                              
      x < 2                               x > 2
      x -3/2.                        x -3/2
Where b) represents the system of the contradictory inequalities.
The red colored part of the graph of the corresponding equilateral or rectangular hyperbola shows the interval (region) where the function's values are negative.
Thus, the solution is    -3/2 < x < 2.
Or, we can transform the given fraction to the quadratic inequality by multiplying it by (2x + 3)2.
So, obtained is     
 (x - 2)(2x + 3) < 0   or    2x2 - x - 6 < 0.
The roots of the corresponding function,  
x - 2 = 0x1 = 2   and   2x + 3 = 0x2 = -3/2
Therefore, the solution is  
-3/2 < x < 2.
The right figure shows that the graph of the function is negative in this interval.
Functions contents A
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