# Sum and Product of the Roots of a Quadratic Equation

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

## Formula 1

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.

## Proof: Formula 1

If *x* = *r*_{1} and *x* = *r*_{2},

then *x* - *r*_{1} = 0 and *x* - *r*_{2} = 0.

Then the quadratic equation

that satisfy these two conditions is

(*x* - *r*_{1})(*x* - *r*_{2}) = 0.

Solving a quadratic equation by factoring

Use the FOIL method

and arrange the trinomial.

Then *x*^{2} - (*r*_{1} + *r*_{2})*x* + *r*_{1}*r*_{2} = 0.

## Example 1

3 + 4 = 7

3⋅4 = 12

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

## Example 2

(2 + *i*) + (2 - *i*) = 4

(2 + *i*)(2 - *i*) = 5

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

## Formula 2

For the quadratic equation *ax*^{2} + *bx* + *c* = 0:*r*_{1} + *r*_{2} = -*b*/*a**r*_{1}*r*_{2} = *c*/*a*

So if the coefficients or the constant term are given,

you can find the sum or the product of the roots.

## Proof: Formula 2

Divide both sides by *a*.

Change the middle term +(*b*/*a*)*x* to -(-*b*/*a*)*x*.

Compare the terms of

this equation and the previous formula:*x*^{2} - (*b*/*a*))*x* + *c*/*a* = 0*x*^{2} - (*r*_{1} + *r*_{2})*x* + *r*_{1}*r*_{2} = 0

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

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

## Example 3

The coefficients of *x*^{2} and *x* are given:

1, -6.

And one of the roots is 2.

Then, if the other root is *r*,

2 + *r* = -6/1.

Then *r* = -8.

## Example 4

The coefficients of *x*^{2} and *x* are given:

2, -1.

And one of the roots is -3.

Then, if the other root is *r*,

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

Then *r* = 7/2.

## Example 5

The coefficient of *x*^{2} and the constant term are given:

3, -12.

And one of the roots is 5.

Then, if the other root is *r*,

5*r* = (-12)/3.

Then *r* = -4/5.