Sum and Product of the Roots of a Quadratic Equation

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 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.

Proof: Formula 1

Sum and Product of the Roots of a Quadratic Equation: Proof of Formula 1

If x = r1 and x = r2,
then x - r1 = 0 and x - r2 = 0.

Then the quadratic equation
that satisfy these two conditions is
(x - r1)(x - r2) = 0.

Solving a quadratic equation by factoring

Use the FOIL method
and arrange the trinomial.

Then x2 - (r1 + r2)x + r1r2 = 0.

Example 1

Write a quadratic equation whose roots are 3 and 4.

3 + 4 = 7
3⋅4 = 12

So the quadratic equation is
x2 - 7x + 12 = 0.

Example 2

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

(2 + i) + (2 - i) = 4
(2 + i)(2 - i) = 5

So the quadratic equation is
x2 - 4x + 5 = 0.

Complex conjugates

Formula 2

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:

r1 + r2 = -b/a
r1r2 = 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

Sum and Product of the Roots of a Quadratic Equation: Proof of 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:
x2 - (b/a))x + c/a = 0
x2 - (r1 + r2)x + r1r2 = 0

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

The coefficients of x2 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

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

The coefficients of x2 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

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

The coefficient of x2 and the constant term are given:
3, -12.

And one of the roots is 5.

Then, if the other root is r,
5r = (-12)/3.

Then r = -4/5.