Formula: Sum & Product of Roots
Relationship between equation and roots
Worksheet on this topic
Sum and Product of Roots worksheet (free 25 question pdf with answer key)

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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.
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 $$ \color{Red}{\frac{c}{a}}$$ .
Example 2
The example below illustrates how this formula applies to the quadratic equation x^{2}  2x  8. Again, both formulasfor the sum and the product boil down to b/a and c/a, respectively.
Practice Problems
Problem 1)
Without solving, find the sum and product of the roots
of the equation:
2x^{2} 3x 2 = 0
Identify the coefficients:
a = 2
b = 3
c = 2
Now, substitute these values into the formulas
Sum of roots 
Product of roots 
$$ \color{Red}{\frac{b}{a} } = \frac{(3)}{2} = \frac{3}{2} $$ 
$$ \color{Red}{ \frac{c}{a} } = \frac{2}{2} = 1$$ 
Problem 2)
Without solving, find the sum & product of the roots
of the following equation:
9x^{2} 8x = 15
First, subtract 15 from both sides so that your equation is in the form
0 = ax^{2} + bx + c
rewritten equation: 9x^{2} 8x  15 = 0
Identify the coefficients:
a = 9
b = 8
c = 15
now, substitute these values into the formulas
Sum of roots 
Product of roots 
$$ \color{Red}{\frac{b}{a} } = \frac{(8)}{9} = \frac{ 8}{9} $$ 
$$ \color{Red}{\frac{c}{a} } = \frac{15}{9} = \frac{5}{3} $$ 
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 (x4) and (x2) 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 
$$ \frac{b}{a} = 6 = \frac{6}{1} $$ 
$$ \frac{c}{a} = 8 = \frac{8}{1} $$ 
Now, we know the values of all 3 coefficients
a = 1
b = 6
c = 8
So our final quadratic equation is
y = 1x^{2} 6x + 8
You can double check your work by foiling the binomials (x 4)(x2) to get the same equation
Problem 4) If one root of the equation below is 3, what is the other root?
x^{2} 5x + k = 0
Write down what you know:
a = 1
b = 5
r_{1} = 3
now, substitute these values into the sum of the roots formula
Sum of roots 
r_{1} + r_{2} = b/a 
3 + r_{2} = (5)/1 
3 + r_{2} = 5 
r_{2} = 2 
Therefore the missing root is 2. We can check our work by foiling the binomials
(x3)(x2) =
x^{2} 5x + 6
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