Sum and Product of the Roots of a Quadratic Equation

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 r1 and r2, then the quadratic equation is x^2 - (r1 + r2)x + r1*r2 = 0.

If the roots of a quadratic equation are r1 and r2,

then the quadratic equation is
x2 - (r1 + r2)x + r1r2 = 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?

Write a quadratic equation whose roots are 3 and 4.

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
x2 - 7x + 12 = 0.

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

Write a quadratic equation whose roots are 2 + i and 2 - i.

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) = 22 + 12

Complex conjugates

22 = 4
12 = 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
x2 - 4x + 5 = 0.

Formula: from a Quadratic Equation to the Roots

For the quadratic equation ax^2 + bx + c = 0, r1 + r2 = -b/a and r1*r2 = c/a.

For the quadratic equation ax2 + bx + c = 0,

if the roots are r1 and r2,

then r1 + r2 = -b/a
and r1r2 = c/a.

Example 3

One root of the given quadratic equation is 2. Find the other root. x^2 + 6x + c = 0

One root is 2.
Set the other root as r.

The coefficient of x2 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 of the given quadratic equation is -3. Find the other root. 2x^2 - x + c = 0

One root is -3.
Set the other root as r.

The coefficient of x2 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 of the given quadratic equation is 5. Find the other root. 3x^2 + bx - 12 = 0

One root is 5.
Set the other root as r.

The coefficient of x2 is 3.
The constant term is -12.

Then 5⋅r = (-12)/3.

(-12)/3 = -4

Divide both sides by 5.

Then r = -4/5.