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The Discriminant in Quadratic Equation
Nature of Quadratic Equation's roots, solutions
The discriminant is a number that can be calculated from any quadratic equation
A quadratic equation is an equation that can be written as
ax ² + bx + c where a ≠ 0
The discriminant in a quadratic equation is found by the following formula
and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation.
discriminant= b² − 4ac
Example of the discriminant
- Quadratic equation = y = 3x² + 9x + 5
- The discriminant = 9 ² − 4 • 3 •5
Nature of the Solutions
| Value of the discriminant |
Type and number of Solutions |
Example of graph |
Positive Discriminant
b² − 4ac > 0
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Two Real Solutions If the discriminant is a perect square the roots are rational. Otherwise, they are irrational.
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Discriminant is Zero
b² − 4ac = 0
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One Real Solution |
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Negative Discriminant
b² − 4ac < 0
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No Real Solutions
Two Imaginary Solutions
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- Example 1
Quadratic Equation: y = x² + 2x + 1
- a = 1
- b = 2
- c = 1
The discriminant for this equation is
2² - 4•1 •1= 4 − 4 = 0
Since the discriminant is zero, there should be 1 real solution to this equation.
Below is a picture representing the graph and one solution of this quadratic equation
Graph of y = x² + 2x + 1
Calculate the discriminant to determine the number and nature of the solutions of the following quadratic equation:
y = x² − 2x + 1
In this quadratic equation,
y = x² − 2x + 1
Using our general formula, the discriminant is
(-2)² − 4•1 •1 = 4 − 4 = 0
Since the discriminant is zero,
we should expect 1 real solution which you can see pictured in the graph below.
Use the discriminant to find out the nature and number of solutions:
y = x² − x − 2
In this quadratic equation,
y = x² − x − 2 and its solution
Calculate the discriminant to determine the nature and number of solutions:
y = x² − 1
In this quadratic equation,
y = x² − 1
Calculate the discriminant to determine the nature and number of solutions:
y = x² + 4x − 5
In this quadratic equation,
y = x² + 4x − 5
Calculate the discriminant to determine the nature and number of solutions:
y = x² + 4x + 5
In this quadratic equation,
y = x² + 4x + 5
Below is a picture of this quadratic's graph
Find the discriminant to determine the nature and number of solutions:
y = x² + 4
y = x² + 4
The solutions are 2i and -2i
Below is a a picture of this equations graph
Find the discriminant to determine the nature and number of solutions:
y = x² + 25
y = x² + 25
The solutions are 5i and -5i
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