The nature of roots of a quadratic equation can be determined by the value of its discriminant (D). See the table below.
First calculate the discriminant by the following formula:
If D equals zero, there are two real and equal roots. In other words, there is only one solution to the quadratic equation. In such a case, the graph of the quadratic equation touches the x-axis at one point.
If D is a negative real number (lesser than zero), then the roots are imaginary or complex numbers. This means that they are of the form `ai + b`, where 'i' is the imaginary number. This is also quite logical, since in the quadratic formula, we take the square root of the discriminant D and hence if D is negative, its square root won't be a real number. The graph of such a quadratic equation does not touch or intersect the x-axis at any point.
Calculate the discriminant:
`D = b^2 - 4ac`
`D = 6^2 - 4(1)(5)`
`D = 16`
Use the table above to determine the nature of the roots:
Since D is greater than zero thus there are two real and distinct roots. Further, 16 is a perfect square. Thus, the roots are rational in nature.
Compare the nature of roots to the actual roots:
Here is a graph of the above equation. It has two x-intercepts, -1 and -5, which are its roots or solutions. Notice that -1 and -5 are a pair of real, distinct and rational numbers. This coincides with our observation above the nature of roots above.
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`D = b^2 - 4ac`If D is a positive real number (greater than zero), then there are two real and distinct roots of the quadratic equation. "real" means that the roots are real numbers and "distinct" means that they are not equal (that is, there are two different roots, not two same roots). In this case, the graph of the equation intersects the x-axis at two distinct points.
If D equals zero, there are two real and equal roots. In other words, there is only one solution to the quadratic equation. In such a case, the graph of the quadratic equation touches the x-axis at one point.
If D is a negative real number (lesser than zero), then the roots are imaginary or complex numbers. This means that they are of the form `ai + b`, where 'i' is the imaginary number. This is also quite logical, since in the quadratic formula, we take the square root of the discriminant D and hence if D is negative, its square root won't be a real number. The graph of such a quadratic equation does not touch or intersect the x-axis at any point.
Example
`x^2 + 6x + 5 = 0`Calculate the discriminant:
`D = b^2 - 4ac`
`D = 6^2 - 4(1)(5)`
`D = 16`
Use the table above to determine the nature of the roots:
Since D is greater than zero thus there are two real and distinct roots. Further, 16 is a perfect square. Thus, the roots are rational in nature.
Compare the nature of roots to the actual roots:
Here is a graph of the above equation. It has two x-intercepts, -1 and -5, which are its roots or solutions. Notice that -1 and -5 are a pair of real, distinct and rational numbers. This coincides with our observation above the nature of roots above.