Solve Quadratic Inequalities Graphically

This is a tutorial on how to solve quadratic inequalities graphically. The quadratic inequalities explored are of the type

a x 2 + b x + c < 0
or
a x 2 + b x + c > 0

Review
An applet plots the graph of y = a x 2 + b x + c and displays part of the graph that is below the x axis (y < 0) in blue and part of the graph that is above the x axis (y > 0) in red. To solve a quadratic inequality you just read the interval corresponding to y < 0 or y > 0 depending on the inequality to solve.

Interactive Tutorials

click on the button above "click here to start" to start the applet and MAXIMIZE the window obtained.

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Example 1 : Solve graphically and analytically the quadratic inequality


- x 2 + 3x + 4 < 0

Solution to Example 1:

Graphical solution: Use the applet to set coefficients a = -1, b = 3 and c = 4 and graph the equation y = - x 2 + 3x + 4. The solution set to the inequality - x 2 + 3x + 4 < 0 correspond to the x coordinates of the points on the graph for which y < 0 BLUE. We have two intervals for which y < 0:

(-∞ , -1) U (4 , +∞)

Analytical solution:

  • Factor the left hand term of the given inequality
    - x 2 + 3x + 4 = (x + 1)(-x + 4)

  • To solve the given inequality, we study the sign of

    (x + 1)(-x + 4)

    on the intervals (-∞ , -1), (-1 , 4) and (4 , +∞) where -1 and 4 are the zeros of (x + 1)(-x + 4).

    let x = -2 a value of x on the interval (-∞ , -1). For this value - x 2 + 3x + 4 = -(-2) 2 + 3(-2) + 4 = -6 is negative. The interval (-∞ , -1) is one solution set. Let x = 0 on the interval (-1 , 4). For this value - x 2 + 3x + 4 = 4 is positive. x = 5 is in the interval (4 , +∞). For this value of x, - x 2 + 3x + 4 = -6 is negative. The interval (4 , +∞) is a solution set for the given inequality.

    Hence the solution set of the inequality is given by the union of all intervals for which - x 2 + 3x + 4 is negative:

    (-∞ , -1)U (4 , +∞)

  • Both the graphical and analytical methods give the same answer.

Example 2 : Solve graphically and analytically the equation


-x 2 + 4x - 5 > 0

Solution to Example 2:

Graphical solution: Use the applet to set coefficients a = -1, b = 4 and c = -5 and graph the equation y = -x 2 + 4x - 5. This inequality has no solutions since the whole graph is below the x axis and therefore y < 0 everywhere.

Analytical solution:

  • -x 2 + 4x - 5 cannot be factored over the real numbers. Therefore -x 2 + 4x - 5 has no zeros and its sign does not change. To find the sign of -x 2 + 4x - 5 you need to evaluate it for one single value of x. Let us evaluate -x 2 + 4x - 5 at x = 0
    -(0) 2 + 4(0) - 5 = - 5.

  • -x 2 + 4x - 5 is always negative hence the inequality -x 2 + 4x - 5 > 0 has no solutions.

Exercises: Solve graphically (using the applet) and analytically the following quadratic inequalities.

1: -x 2 - 4 x < -5

2: x 2 - 2 x + 8 >= 0

3: x 2 - 3 x <= 0

Solutions to Above Exercises:

1: (-∞ , -5) U (1 , +∞)

2: (-∞ , +∞)

3: [0 , 3]

More references and links on how to Solve Equations, Systems of Equations and Inequalities.

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