# Quadratic Equation: Sum and Product of the Roots

How to find the quadratic equation from the sum and product of the roots (and vice versa): 2 formulas, 4 examples, and their solutions.

## From Roots to Quadratic Equation

### Formula

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 you know the roots,

you can write the quadratic equation.

## Example 1

### Example

### Solution

The roots are 3 and 4.

Then the quadratic equation is

x^{2} - (3 + 4)x + 3⋅4 = 0.

-(3 + 4) = -7

+3⋅4 = +12

So

x^{2} - 7x + 12 = 0

is the answer.

## Example 2

### Example

### Solution

If one root is 2 + i,

then the other root is,

the conjugate of 2 + i,

2 - i.

This is true because

the imaginary number part comes from

±√b^{2} - 4ac.

And this ± makes the conjugate roots.

Quadratic Equation: Complex Roots

The roots are 2 + i and 2 - i.

Then the quadratic equation is

x^{2} - [(2 + i) + (2 - i)]x + (2 + i)(2 - i) = 0.

-[(2 + i) + (2 - i)]

= -[2 + 2]

= -4

Add and Subtract Complex Numbers

+(2 + i)(2 - i)

= +2^{2} + 1^{2}

Divide Complex Numbers: Formula

+2^{2} + 1^{2} = +4 + 1

+4 + 1 = +5

So

x^{2} - 4x + 5 = 0

is the answer.

## From Quadratic Equation to Sum and Product of the Roots

### Formula

For a quadratic equation

ax^{2} + bx + c = 0

(a ≠ 0),

r_{1} + r_{2} = -a/c

r_{1}r_{2} = b/c

So, from a quadratic equation,

you can find

the sum and the product of the roots.

## Example 3

### Example

### Solution

It says

one root is 2.

Then set the other root r.

So the roots are

2 and r.

The given quadratic equation is

1x^{2} + 6x + c = 0.

The x^{2} term and the x term are known.

The roots are 2 and r.

Then

r + 2 = -6/1.

-6/1 = -6

Move +2 to the right side.

Then r = -8.

So the other root is -8.

## Example 4

### Example

### Solution

It says

one root is 5.

Then set the other root r.

So the roots are

5 and r.

The given quadratic equation is

3x^{2} + bx - 15 = 0.

The x^{2} term and the constant term are known.

The roots are 5 and r.

Then

5⋅r = -15/3.

-15/3 = -5

Divide both sides by 5

Then r = -1.

So the other root is -1.