**Sum and Product of Roots of Quadratic Equation : **

In this section, you will learn how to find sum and product of the roots of a quadratic equation easily.

If a quadratic equation is given in standard form, we can find the sum and product of the roots using coefficient of x^{2}, x and constant term.

Let us consider the standard form of a quadratic equation,

ax^{2} + bx + c = 0

(Here a, b and c are real and rational numbers)

Let α and β be the two zeros of the above quadratic equation.

Then the formula to get sum and product of the roots of a quadratic equation is,

**Example 1 :**

Find the sum and product of roots of the quadratic equation given below.

x^{2} - 5x + 6 = 0

**Solution :**

Comparing

x^{2} - 5x + 6 = 0

and

ax^{2} + bx + c = 0

we get

a = 1, b = -5 and c = 6

Therefore,

Sum of the roots = -b/a = -(-5)/1 = 5

Product of the roots = c/a = 6/1 = 6

**Example 2 :**

Find the sum and product of roots of the quadratic equation given below.

x^{2} - 6 = 0

**Solution :**

Comparing

x^{2} - 6 = 0

and

ax^{2} + bx + c = 0

we get

a = 1, b = 0 and c = -6

Therefore,

Sum of the roots = -b/a = 0/1 = 0

Product of the roots = c/a = -6/1 = -6

**Example 3 :**

Find the sum and product of roots of the quadratic equation given below.

3x^{2} + x + 1 = 0

**Solution :**

Comparing

3x^{2} + x + 1 = 0

and

ax^{2} + bx + c = 0

we get

a = 3, b = 1 and c = 1

Therefore,

Sum of the roots = -b/a = -1/3

Product of the roots = c/a = 1/3

**Example 4 :**

Find the sum and product of roots of the quadratic equation given below.

3x^{2} + 7x = 2x - 5

**Solution :**

First write the given quadratic equation in standard form.

3x^{2} +7x = 2x - 5

3x^{2} + 5x + 5 = 0

Comparing

3x^{2} + 5x + 5 = 0

and

ax^{2} + bx + c = 0

we get

a = 3, b = 5 and c = 5

Therefore,

Sum of the roots = -b/a = -5/3

Product of the roots = c/a = 5/3

**Example 5 :**

Find the sum and product of roots of the quadratic equation given below.

3x^{2} -7x + 6 = 6

**Solution :**

First write the given quadratic equation in standard form.

3x^{2} -7x + 6 = 6

3x^{2} - 7x = 0

Comparing

3x^{2} - 7x = 0

and

ax^{2} + bx + c = 0

we get

a = 3, b = -7 and c = 0

Therefore,

Sum of the roots = -b/a = -(-7)/3 = 7/3

Product of the roots = c/a = 0/3 = 0

**Example 6 :**

Find the sum and product of roots of the quadratic equation given below.

x^{2} + 5x + 1 = 3x^{2} + 6

**Solution :**

First write the given quadratic equation in standard form.

x^{2} + 5x + 1 = 3x² + 6

0 = 2x^{2} - 5x + 5

2x^{2} - 5x + 5 = 0

Comparing

2x^{2} - 5x + 5 = 0

and

ax^{2} + bx + c = 0

we get

a = 2, b = -5 and c = 5

Therefore,

Sum of the roots = -b/a = -(-5)/3 = 5/2

Product of the roots = c/a = 5/2

After having gone through the stuff given above, we hope that the students would have understood how find the sum and product of the roots of a quadratic equation without solving the equation.

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