Roots of a Quadratic Equation and Their Nature

🎯 What will you learn in this lesson?

👉 What the roots of a quadratic equation mean,

👉 how to find roots by using Sridharacharya's formula,

👉 the sum and product of the roots,

👉 how to form a quadratic equation from its roots,

👉 how to use the discriminant D = b² − 4ac to know the nature of roots.

What is a Root of a Quadratic Equation?

⭐ The general form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0 and a, b, c are constants.

The value of x that makes the equation true is called a root of that equation.

📌 In general, a quadratic equation can have at most two roots. The two roots may be different, or they may be equal.

✍ Example: Checking the roots

Consider the equation x² − 5x + 6 = 0.

Put x = 2 in the left side:

2² − 5 × 2 + 6 = 4 − 10 + 6 = 0

So, 2 is a root.

Now put x = 3:

3² − 5 × 3 + 6 = 9 − 15 + 6 = 0

So, 3 is also a root.

Therefore, the two roots of x² − 5x + 6 = 0 are 2 and 3.

Simple idea

A root is like a correct answer for x. When we put that value in the equation, the equation becomes true.

Finding Roots by Sridharacharya's Formula

For the quadratic equation ax² + bx + c = 0, where a ≠ 0, the roots are found by the formula:

👉

Start with the general form:

ax² + bx + c = 0

Since a ≠ 0, divide every term by a:

Now complete the square:

Taking square roots on both sides:

Therefore,

This formula is commonly known in India as Sridharacharya's Formula. It is also called the quadratic formula.

Here, D = b² − 4ac is called the discriminant.

✍ Example: Finding roots using the formula

Find the roots of 2x² − 7x + 3 = 0.

Here, a = 2, b = −7, and c = 3.

D = b² − 4ac = (−7)² − 4 × 2 × 3 = 49 − 24 = 25

Using Sridharacharya's formula,

So,

and

Therefore, the roots are 3 and .

Sum and Product of the Roots

Let α and β be the roots of the quadratic equation ax² + bx + c = 0.

Then, and

👉 Why are these formulas true?+

From Sridharacharya's formula, the two roots are:

and

Now add them:

The square-root terms cancel each other.

Now multiply them:

Using (p + q)(p − q) = p² − q²,

✍ Example: Finding the sum and product of roots

For 3x² − 10x + 3 = 0, let the roots be α and β.

Here, a = 3, b = −10, and c = 3.

So, the sum of the roots is , and the product of the roots is 1.

✍ Simple practice example

For x² − 8x + 15 = 0,

a = 1, b = −8, c = 15.

Sum of roots

Product of roots

Forming a Quadratic Equation from Given Roots

If the two roots are α and β, then the quadratic equation is:

In words: x² − (sum of the roots)x + (product of the roots) = 0

👉 How do we get this formula?+

If α and β are the roots, then:

and

Divide ax² + bx + c = 0 by a:

Since and , we get:

✍ Example: Form an equation from integer roots

Form a quadratic equation whose roots are 4 and −1.

Sum of roots = 4 + (−1) = 3

Product of roots = 4 × (−1) = −4

Therefore, the equation is:

x² − 3x − 4 = 0

✍ Example: Form an equation from fractional roots

Form a quadratic equation whose roots are and 3.

So, the equation is:

To remove fractions, multiply the whole equation by 2:

2x² − 7x + 3 = 0

Nature of the Roots of a Quadratic Equation

In the formula

the expression under the square root, b² − 4ac, decides the nature of the roots.

So, D = b² − 4ac is called the discriminant.

Main cases of the discriminant

Value of discriminant	Nature of roots
D > 0	The roots are real and unequal.
D = 0	The roots are real and equal.
D < 0	There are no real roots. In higher mathematics, the roots are imaginary or complex.
D > 0 and D is a perfect square	The roots are real, rational, and unequal.
D > 0 but D is not a perfect square	The roots are real, irrational, and unequal.

Important points to remember

📚 The roots are real when D ≥ 0.

📚 If a quadratic equation has rational coefficients, then one root cannot be rational and the other irrational. Either both are rational, or both are irrational.

📚 If a quadratic equation has real coefficients, then one root cannot be real and the other imaginary. Either both roots are real, or both roots are imaginary or complex.

📚 For Class 10, when D < 0, you can simply write: no real roots.

✍ Example: D > 0 and D is a perfect square

Consider x² − 5x + 6 = 0.

Here, a = 1, b = −5, and c = 6.

D = b² − 4ac = 25 − 24 = 1

Since D = 1 > 0 and 1 is a perfect square, the roots are real, rational, and unequal.

The roots are 2 and 3.

✍ Example: D > 0 but D is not a perfect square

Consider x² − 2x − 1 = 0.

Here, a = 1, b = −2, and c = −1.

D = b² − 4ac = 4 − 4 × 1 × (−1) = 8

Since D = 8 > 0, the roots are real and unequal. But 8 is not a perfect square, so the roots are irrational.

✍ Example: D = 0

Consider x² − 6x + 9 = 0.

Here, a = 1, b = −6, and c = 9.

D = b² − 4ac = 36 − 36 = 0

So, the roots are real and equal. Both roots are 3.

✍ Example: D < 0

Consider x² + 4x + 5 = 0.

Here, a = 1, b = 4, and c = 5.

D = b² − 4ac = 16 − 20 = −4

So, there are no real roots.

In higher mathematics, the roots are −2 + i and −2 − i.

Some Important Theorems on Quadratic Equations

💡 If α is a root of the quadratic equation ax² + bx + c = 0, then (x − α) is a factor of that equation. Similarly, if (x − α) is a factor, then α is a root.

💡 A quadratic equation cannot have more than two roots.

💡 If a quadratic equation has rational coefficients and one root is irrational, then the other root will be its irrational conjugate. For example, if one root is , then the other root is .

💡 If a quadratic equation has real coefficients and one root is imaginary, then the other root will be its imaginary conjugate. For example, if one root is p + iq, then the other root is p − iq, where .

Exam Points to Remember

The general form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
The formula for roots is .
The discriminant is D = b² − 4ac.
If D > 0, the roots are real and unequal.
If D = 0, the roots are real and equal.
If D < 0, there are no real roots.
Sum of roots:
Product of roots:
If the roots are α and β, then the equation is x² − (α + β)x + αβ = 0.

💡 Common Mistakes

Common mistake	Correct idea
Thinking that the equation is quadratic even when a = 0.	If a = 0, the x² term is absent. So the equation is not quadratic.
Thinking that D > 0 always means rational roots.	Roots are rational only when D > 0 and D is a perfect square.
Writing that there are no roots when D < 0.	There are no real roots. In higher mathematics, imaginary or complex roots exist.
Writing the sum of roots as .	The correct sum is .
Making sign mistakes while forming an equation from roots.	Use the formula x² − (sum of roots)x + (product of roots) = 0.

Quick Revision

A root is a value of x that satisfies the equation.
A quadratic equation has at most two roots.
Sridharacharya's formula helps us find the roots directly.
The discriminant D = b² − 4ac tells us the nature of roots.
Sum of roots is .
Product of roots is .
Equation from roots: x² − (sum)x + (product) = 0.