Nature of roots — lesson. Mathematics State Board, Class 10.

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We have learnt that the roots of the quadratic equation \(ax^2 + bx + c = 0\) can be found by the quadratic formula:

x = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}

Important!

\(b^2 - 4ac\) is called the discriminant of the quadratic equation \(ax^2 + bx + c = 0\). It is denoted by the letter \(\Delta\) or \(D\).

Let us discuss the nature of the roots of the quadratic equation depending on the discriminant.

Case I: \(\Delta = b^2 - 4ac > 0\)

Here, \(b^2 - 4ac > 0\). That means the value of the discriminant is positive.

Then, the possible roots are

\frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}

and

\frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}

If \(\Delta = b^2 - 4ac > 0\) then the roots are real and distinct.

Case II: \(\Delta = b^2 - 4ac = 0\)

Here, \(b^2 - 4ac = 0\). That means the value of the discriminant is zero.

x = \frac{- b + \sqrt{0}}{2 a}

and

x = \frac{- b - \sqrt{0}}{2 a}

x = \frac{- b}{2 a}

and

x = \frac{- b}{2 a}

The possible roots are

\frac{- b}{2 a}

and

\frac{- b}{2 a}

If \(\Delta = b^2 - 4ac = 0\), then the roots are real and equal.

Case III: \(\Delta = b^2 - 4ac < 0\)

Here, \(b^2 - 4ac < 0\). That means the value of the discriminant is negative.

We won't get any real roots in this case.

If \(\Delta = b^2 - 4ac < 0\) then there are no real roots.

Relation between roots and coefficients of a quadratic equation

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then:

(i) Sum of the roots \(=\)

α + β = \frac{- b}{a}

(ii) Product of the roots \(=\)

α β = \frac{c}{a}

Quadratic equation \(=\) \(x^2 - (\text{sum of the roots})x + \text{product of the roots}\)

Important!

To know more about the relationship between roots and coefficients of a equation click here.

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1. Nature of roots