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Example 1 : Find all values of the parameter m in the quadratic equation
x 2 + mx + 1 = 0
- one solution,
- 2 real solutions, and
- 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.
- 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.
- 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)
Matched Exercise 1: Find all values of the parameter m in the quadratic equation
x 2 + x + m + 1 = 0
- one solution,
- 2 real solutions, and
- 2 complex solutions.
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.