Solve Quadratic Equations Using Discriminants (2)

This is a tutorial on using the discriminant and the quadratic formula to determine the number and nature of the solutions to the quadratic equations. Detailed solutions and explanations are included. This is a continuation of tutorial (1) on quadratic equations. Exercises with answers are at the bottom of this page.



Example 1 : Find all values of the parameter m in the quadratic equation


x 2 + mx + 1 = 0
such that the equation has
  1. one solution,
  2. 2 real solutions, and
  3. 2 complex solutions.

Solution to Example 1:

  • Given
    x 2 + mx + 1 = 0

  • Find the discriminant D = b2 - 4ac
    D = b2 - 4ac = m2 - 4(1)(1) = m2 - 4

  • For the equation to have one solution, the discriminant has to be equal to zero.
    m2 - 4 = 0

  • The equation m2 - 4 = 0 has two solutions.
    m = 2
    m = -2

    Below is the graph of the expression in the left side of the given equation for m = 2 and m = -2. Note that in each case, the graph has 1 x intercept only, hence one real solution to the equation.
    graphical solution of the given quadratic equation  for m = -2 and m = 2.

  • For the equation to have 2 real solution, the discriminant has to be greater than zero.
    m2 - 4 > 0

  • The inequality m2 - 4 > 0 has the following solution set.
    (-infinity , -2) U (2 , +infinity)

    Below is the graph of the expression in the left side of the given equation for m = 5 and m = -3. Note that in each case, the graph has 2 x intercepts, hence 2 real solutions to the equation.
    graphical solution of the given quadratic equation  for m = -2 and m = 2.

  • For the equation to have 2 complex solution, the discriminant has to be less than zero.
    m2 - 4 < 0

  • The inequality m2 - 4 < 0 has the following solution set.
    (-2 , 2)


Below is the graph of the expression in the left side of the given equation for m = 0 and m = 1. Note that in each case, the graph has no x intercepts, hence the solutions to the equation are not real but complex.
graphical solution of the given quadratic equation  for m = 0 and m = 1.

Matched Exercise 1: Find all values of the parameter m in the quadratic equation


x 2 + x + m + 1 = 0
such that the equation has
  1. one solution,
  2. 2 real solutions, and
  3. 2 complex solutions.

Detailed Solution

Exercises.(see answers below)

For what value of m the following quadratic equation has no solutions?

a) 2x 2 + mx + 2 = 0

For what value of m the following quadratic equation has two solutions?

b) x 2 + (1/m) x = -1

For what value of m the following quadratic equation has one solution?

c) x 2 + m = 0

Answers to Above Exercises.

a) m in the interval (-4 , 4)

b) m in the intervals (-1/2 , 0) U (0 , 1/2)

c) m = 0

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

Tutorial on Equations of the Quadratic Form.

Equations with Rational Expressions - Tutorial.


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Updated: 3 April 2011

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