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Formula: Sum & Product of Roots

Relationship between equation and roots

Worksheet on this topic
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Sum and Product of Roots worksheet (free 25 question pdf with answer key)
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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
Answer


Problem 2) Without solving, find the sum & product of the roots of the following equation:
-9x2 -8x = 15
Answer


Problem 3) Write the quadratic equation given the following roots:
4 and 2
Answer


Problem 4) If one root of the equation below is 3, what is the other root?
x2 -5x + k = 0
Answer
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