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

**Practice** Problems

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**

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

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**

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

**Product of roots**

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

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 = 1x^{2} - 6x + 8

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

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

(x-3)(x-2) = x^{2} -5x + 6