Formula: Sum & Product of Roots

Relationship between equation and roots

The formulas

sum of roots: $$ \frac{ -b}{a} $$
product of roots: $$ \frac{ c}{a} $$

As you can see from the work below, when you are trying to solve a quadratic equations in the form of $$ ax^2 +bx + c$$. The sum and product of the roots can be rewritten using the two formulas above.

Sum and Product Formula Derived
Example 1

The example below illustrates how this formula applies to the quadratic equation $$ x^2 + 5x +6 $$. As you, can see the sum of the roots is indeed $$\color{Red}{ \frac{-b}{a}}$$ and the product of the roots is $$ \color{Red}{\frac{c}{a}}$$ .

Picture of sum and product of roots formula
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.

Picture of sum and product of roots formula

Practice Problems

Problem 1

Without solving, find the sum and product of the roots of the equation: 2x2 -3x -2 = 0

Identify the coefficients:
a = 2
b = -3
c = -2

Now, substitute these values into the formulas

Sum of roots

$$ \color{Red}{\frac{-b}{a} } = \frac{-(-3)}{2} = \frac{3}{2} $$

Product of roots

$$ \color{Red}{ \frac{c}{a} } = \frac{-2}{2} = -1 $$

Problem 2

Without solving, find the sum & product of the roots of the following equation: -9x2 -8x = 15

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, substitute these values into the formulas

Sum of roots

$$ \color{Red}{\frac{-b}{a} } = \frac{-(-8)}{-9} = \frac{ -8}{9} $$

Product of roots

$$ \color{Red}{\frac{c}{a} } = \frac{-15}{9} = \frac{-5}{3} $$

Problem 3

Write the quadratic equation given the following roots: 4 and 2

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

$$ \frac{-b}{a} = 6 = \frac{6}{1} $$

Product of roots

$$ \frac{c}{a} = 8 = \frac{8}{1} $$

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 + k = 0

Write down what you know:
a = 1
b = -5
r1 = 3

Now, substitute these values into the sum of the roots formula

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

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