1. Examples
If `alpha` and `beta` are the roots of the quadratic equation `ax^2+bx+c=0`
then `alpha=(-b+sqrt(Delta))/(2a)` and `beta=(-b-sqrt(Delta))/(2a)` where `Delta=b^2-4ac`
`Delta` is called the discriminant and read as 'delta'
`Delta` and nature of the roots:
1. If `Delta` > 0 then the roots are real and distinct.
(i) If `Delta` is a perfect square then the roots are rational and distinct.
(ii) If `Delta` is not a perfect square then the roots are irrational and distinct.
2. If `Delta` = 0 then the roots are real and equal.
3. If `Delta` < 0 then the quadratic equation has no real roots.
Example :
1. Find the discriminant of Quadratic Equation `x^2-5x+6=0` and discuss the nature of its roots
Solution:
`x^2-5x+6=0`
`=>x^2-5x+6 = 0`
Comparing the given equation with the standard quadratic equation `ax^2+bx+c=0,`
we get, `a=1, b=-5, c=6.`
`:. Delta=b^2-4ac`
`=(-5)^2-4 (1) (6)`
`=25-24`
`=1`
`=(1)^2`
Here, `Delta > 0` and is a perfect square. Also a and b are rational.
Hence, the roots of the equation are unequal(distinct) and rational.
2. Find the discriminant of Quadratic Equation `2x^2+5x-10=0` and discuss the nature of its roots
Solution:
`2x^2+5x-10=0`
`=>2x^2+5x-10 = 0`
Comparing the given equation with the standard quadratic equation `ax^2+bx+c=0,`
we get, `a=2, b=5, c=-10.`
`:. Delta=b^2-4ac`
`=(5)^2-4 (2) (-10)`
`=25+80`
`=105`
Here, `Delta > 0` but not a perfect square.
Hence, the roots of the equation are unequal(distinct) and irrational.
3. Find the discriminant of Quadratic Equation `9x^2-24x+16=0` and discuss the nature of its roots
Solution:
`9x^2-24x+16=0`
`=>9x^2-24x+16 = 0`
Comparing the given equation with the standard quadratic equation `ax^2+bx+c=0,`
we get, `a=9, b=-24, c=16.`
`:. Delta=b^2-4ac`
`=(-24)^2-4 (9) (16)`
`=576-576`
`=0`
Here, `Delta=0,` the roots of the equation are real and equal.
and since a and b are both rational, the roots are rational.
Thus, the roots of the given equation are equal and rational.
4. Find the discriminant of Quadratic Equation `4x^2+11x+10=0` and discuss the nature of its roots
Solution:
`4x^2+11x+10=0`
`=>4x^2+11x+10 = 0`
Comparing the given equation with the standard quadratic equation `ax^2+bx+c=0,`
we get, `a=4, b=11, c=10.`
`:. Delta=b^2-4ac`
`=(11)^2-4 (4) (10)`
`=121-160`
`=-39`
Here, `Delta < 0`
Hence, the equation has no real roots.
This material is intended as a summary. Use your textbook for detail explanation.
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