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. Also included a Step by Step Quadratic Equation Solver.

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.

Step by Step Quadratic Equation Solver.

Tutorial on Equations of the Quadratic Form.

Equations with Rational Expressions - Tutorial.