﻿ Formula for the Sum and Product of the Roots of a Quadratic Equation # 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.

##### 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 x2 - 2x - 8. Again, both formulas - for the sum and the product boil down to -b/a and c/a, respectively.

### Practice Problems

##### Problem 1

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

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

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

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

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