Roots of Quadratic Equation

Here you will learn what are the roots of quadratic equation with examples.

Let’s begin – 

Roots of Quadratic Equation

If p(x) = 0 is a quadratic equation, then the zeroes of the poynomial p(x) are called the roots of the quadratic equation p(x) = 0.

Thus, x = 1 is a root of p(x) = \(x^2 + x – 2\) = 0

Since p(1) = \((1)^2\) + 1 – 2 = 0

and, x = -2 is also a root of \(x^2 + x – 2\) = 0.

Since p(-2) = \((-2)^2\) + (-2) – 2 = 4 – 2 – 2 = 0.

Note

1). x = \(\alpha\) is a root of p(x) = 0, iff \(p(\alpha)\) = 0.

2). Every quadratic equation can have atmost two real roots.

3). Finding the zeroes of the quadratic equations is known as solving the quadratic equation.

Example : In each of the following quadratic equations the values of x are given. Determine whether these values are roots of the equations or not.

(i) \(2x^2 + 3x + 1\) = 0;  x = -1

(ii)  \(4x^2 – 5x – 6\) = 0;  x = 2

Solution

(i)  Putting x = -1, on the left side of the given equation, we get

L.H.S = \(2\times (-1)^2 + 3\times (-1) + 1\) = 2 – 3 + 1 = 0 = R.H.S

So, x = -1 is a root of the given equation.

(ii)  Putting x = 2, on the left side of the given equation, we get

L.H.S = \(4\times (2)^2 – 5\times (2) – 6\) = 16 – 10 – 6 = 0 = R.H.S

So, x = 2 is a root of the given equation.

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