In the previous section, you have seen that the roots of the equation `ax^2 + bx + c = 0` are given by
`" " color(blue)(x = (-b pm sqrt(b^2-4ac))/(2a))`
`color(red)("Case 1 : ")` If `color(red)(b^2 – 4ac > 0),` we get two distinct real roots `color(blue)(- b/(2a) + sqrt(b^2-4ac)/(2a))` and `color(blue)(-b/(2a) - sqrt(b^2-4ac)/(2a))`
`color(red)("Case 2 : ")` If `color(red)(b^2 – 4ac = 0),` then `color(blue)(x = -b/(2a) pm 0)` i.e. `x = -b/(2a)` or `-b/(2a)`
So, the roots of the equation `ax^2 + bx + c = 0` are both `color(blue)((-b)/(2a))`
Therefore, we say that the quadratic equation `ax^2 + bx + c = 0` has two equal real roots in this case.
`color(red)("Case 3 : ")` If `color(red)(b^2 – 4ac < 0,)` then there is no real number whose square is `b^2 – 4ac.`
Therefore, there are no real roots for the given quadratic equation in this case.
`color(green)(ul★"Discriminant of this Quadratic Equation")`
Since `color(red)(b^2 – 4ac)` determines whether the quadratic equation `ax^2 + bx + c = 0` has real roots or not, `b^2 – 4ac` is called `color(blue)("the discriminant of this quadratic equation.")`
So, a quadratic equation `color(red)(ax^2 + bx + c = 0)` has
`color(red)((i))` two distinct real roots, if `color(blue)(b^2 – 4ac > 0),`
`color(red)((ii))` two equal real roots, if `color(blue)(b^2 – 4ac = 0),`
`color(red)((iii))` no real roots, if `color(blue)(b^2 – 4ac < 0).`
In the previous section, you have seen that the roots of the equation `ax^2 + bx + c = 0` are given by
`" " color(blue)(x = (-b pm sqrt(b^2-4ac))/(2a))`
`color(red)("Case 1 : ")` If `color(red)(b^2 – 4ac > 0),` we get two distinct real roots `color(blue)(- b/(2a) + sqrt(b^2-4ac)/(2a))` and `color(blue)(-b/(2a) - sqrt(b^2-4ac)/(2a))`
`color(red)("Case 2 : ")` If `color(red)(b^2 – 4ac = 0),` then `color(blue)(x = -b/(2a) pm 0)` i.e. `x = -b/(2a)` or `-b/(2a)`
So, the roots of the equation `ax^2 + bx + c = 0` are both `color(blue)((-b)/(2a))`
Therefore, we say that the quadratic equation `ax^2 + bx + c = 0` has two equal real roots in this case.
`color(red)("Case 3 : ")` If `color(red)(b^2 – 4ac < 0,)` then there is no real number whose square is `b^2 – 4ac.`
Therefore, there are no real roots for the given quadratic equation in this case.
`color(green)(ul★"Discriminant of this Quadratic Equation")`
Since `color(red)(b^2 – 4ac)` determines whether the quadratic equation `ax^2 + bx + c = 0` has real roots or not, `b^2 – 4ac` is called `color(blue)("the discriminant of this quadratic equation.")`
So, a quadratic equation `color(red)(ax^2 + bx + c = 0)` has
`color(red)((i))` two distinct real roots, if `color(blue)(b^2 – 4ac > 0),`
`color(red)((ii))` two equal real roots, if `color(blue)(b^2 – 4ac = 0),`
`color(red)((iii))` no real roots, if `color(blue)(b^2 – 4ac < 0).`