Formula: Sum & Product of Roots

The formulas

sum of roots: −b/a
product of roots: c/a

As you can see from the derivation below, when you are trying to solve aquadratic equations in the form of ax²+bx +c. The sum and product of the roots can be rewritten using the two formulas above.

Example 1

The example below illustrates how this formula applies to the quadratic equation x² + 5x +6. As you can see the sum of the roots is indeed -b/a and the product of the roots is c/a.

Example 2

The example below illustrates how this formula applies to the quadratic equation x² - 2x - 8. Again, both formulas--for the sum and the product boil down to -b/a and c/a, respectively.

Problem 1) Without solving, find the sum and product of the roots equation:

2x² -3x -2 = 0

Answer

Identify the coefficients:
a = 2
b = -3
c = -2
now, subsitute these values into the formulas

Sum of roots	Product of roots
-b/a = -(-3)/2 = 3/2	c/a = -2/2 = -1

Problem 2) Without solving, find the sum & product of the roots of the following equation:

-9x² -8x = 15

Answer

First, subtract 15 from both sides so that your equation is in the form

0 = ax² + bx + c

rewritten equation: -9x² -8x - 15 = 0
Identify the coefficients:
a = -9
b = -8
c = -15
now, subsitute these values into the formulas

Sum of roots	Product of roots
-b/a = -(-8)/-9 = -8/9	c/a = -15/9 = -5/3

Problem 3) Write the quadratic equation given the following roots:

4 and 2

Answer

There are a few ways to approach this kind of problem, you could create two binomials (x-4) and (x-2) and multiply them.

However, since this page focuses using our formulas, let's use them to answer this equation.

Sum of the roots = 4 + 2 = 6
Product of the roots = 4 * 2 = 8

We can use our formulas, to set up the following two equations

Sum of roots	Product of roots
-b/a = 6 = 6/1	c/a = 8 = 8/1

Now, we know the values of all 3 coefficients
a = 1
b = -6
c = 8
So our final quadratic equation is

y = 1x² -6x + 8

You can double check your work by foiling the binomials (x -4)(x-2) to get the same equation

Problem 4)If one root of the equation below is 3, what is the other root?

x² -5x + 6 = 0

Answer

Write down what you know:
a = 1
b = -5
r₁ = 3
now, subsitute these values into the sum of the roots ormula

Sum of roots

r1 + r2 = -b/a

3 + r2 = -(-5)/1

3 + r2 = 5

r2 = 2

Therefore the missing root is 2. We can check our work by foiling the binomials
(x-3)(x-2) = x² -5x + 6