Nature of Roots (Sum and Product)
Solving quadratic equations by factoring, such as the example at the right, is a well honed skill at this point in your mathematical career.
But did you ever stop to notice how the roots of equations are related to the coefficients and constants of the equation itself?
Let’s investigate:
Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term.
The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient.
(r1+r2) = -b/a
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
r1r2 = c/a
You will discover, as you progress in your mathematical career, that these types of relationships also extend to equations of higher degree.
Example: Write a quadratic equation whose roots are -3 and 1/2
Of course, this question could be answered by simply multiplying the factors formed by these roots:
(x+3).(x-1/2) = 0
But with our new found discoveries, we can also arrive at the answer by utilizing the relationship between the roots and coefficients and constants.
