Solve Quadratic Equations Using Discriminants (2)

A tutorial on using the discriminant and the quadratic formula to determine the number and nature of the solutions to the quadratic equations. Questions with detailed solutions and explanations are included. This is a continuation of tutorial (1) on quadratic equations. More questions with answers are at the bottom of this page. Also included in this website, a Step by Step Quadratic Equation Solver.

Question 1 :

Find all values of the parameter m in the quadratic equation

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

Solution to Question 1:

  • a) Given
    x 2 + m x + 1 = 0
  • Find the discriminant Δ = b2 - 4ac
    Δ = 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.

  • b) 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.

  • c) 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 Question 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

More Questions.

(see answers below)
For what value of m the following quadratic equation has no real solutions?
a) 2x
2 + mx + 2 = 0
For what value of m the following quadratic equation has two real 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 Questions.


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

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.