# Sum and Product of the Roots of a Quadratic Equation

How to use the sum and product of the roots of a quadratic equation: formulas, examples, and their solutions.

## Formula: from the Roots to a Quadratic Equation

If the roots of a quadratic equation are *r*_{1} and *r*_{2},

then the quadratic equation is*x*^{2} - (*r*_{1} + *r*_{2})*x* + *r*_{1}*r*_{2} = 0.

So if the roots of a quadratic equation are given,

then you can directly write the quadratic equation.

## Example 1: Roots 3 and 4, Quadratic Equation?

It says

the roots are 3 and 4.

So its sum is, 3 + 4, 7.

And its product is, 3⋅4, 12.

The sum of the roots is 7.

The product of the roots is 12.

So the quadratic equation is*x*^{2} - 7*x* + 12 = 0.

## Example 2: Roots 2 + *i* and 2 - *i*, Quadratic Equation?

It says

one root is 2 + *i*.

Then the other root is 2 - *i*,

because the *i*

comes from the [± term] of the quadratic formula.

Complex roots of a quadratic equation

The sum of the roots is

(2 + *i*) + (2 - *i*).

Cancel +*i* and -*i*.

Then the sum is, 2 + 2, 4.

The product of the roots is (2 + *i*)(2 - *i*).

(2 + *i*)(2 - *i*) = 2^{2} + 1^{2}

Complex conjugates

2^{2} = 4

1^{2} = 1

4 + 1 = 5

So the product of the roots is 5.

The sum of the roots is 4.

The product of the roots is 5.

So the quadratic equation is*x*^{2} - 4*x* + 5 = 0.

## Formula: from a Quadratic Equation to the Roots

For the quadratic equation *ax*^{2} + *bx* + *c* = 0,

if the roots are *r*_{1} and *r*_{2},

then *r*_{1} + *r*_{2} = -*b*/*a*

and *r*_{1}*r*_{2} = *c*/*a*.

## Example 3

One root is 2.

Set the other root as *r*.

The coefficient of *x*^{2} is 1.

The coefficient of *x* is +6.

Then 2 + *r* = -6/1.

-6/1 = -6

Move 2 to the right side.

Then *r* = -8.

This is the other root.

## Example 4

One root is -3.

Set the other root as *r*.

The coefficient of *x*^{2} is 2.

The coefficient of *x* is -1.

Then -3 + *r* = -(-1)/2.

-(-1)/2 = 1/2

Move -3 to the right side.

Then *r* = 1/2 + 3.

To add 1/2 and 3,

multiply 2/2 to +3.

3⋅2 = 6

1 + 6 = 7

So *r* = 7/2.

## Example 5

One root is 5.

Set the other root as *r*.

The coefficient of *x*^{2} is 3.

The constant term is -12.

Then 5⋅*r* = (-12)/3.

(-12)/3 = -4

Divide both sides by 5.

Then *r* = -4/5.