Quadratic Polynomial with Sum and Product of Zeroes

To find a quadratic polynomial given the sum and product of its zeroes, we can use the fact that the sum of the zeroes is equal to the opposite of the coefficient of the linear term, and the product of the zeroes is equal to the constant term divided by the coefficient of the quadratic term.

Given:
Sum of zeroes = -5
Product of zeroes = 6

Let's denote the two zeroes as a and b. We can write the quadratic polynomial in the standard form as:
f(x) = ax^2 + bx + c

Sum of zeroes:
The sum of the zeroes is equal to -b/a. Therefore, we have:
a + b = -5

Product of zeroes:
The product of the zeroes is equal to c/a. Therefore, we have:
ab = 6

We can use these two equations to find the values of a and b.



Solving the Equations

We can solve these equations by substitution or elimination method. Let's use the substitution method.

From the first equation, we have:
a = -5 - b

Substituting this value of a into the second equation, we get:
(-5 - b)b = 6

Expanding the equation:
-5b - b^2 = 6

Rearranging the equation:
b^2 + 5b + 6 = 0

We now have a quadratic equation in terms of b. Factoring the equation, we get:
(b + 2)(b + 3) = 0

Setting each factor equal to zero, we have two possible values for b:
b + 2 = 0 or b + 3 = 0

Solving these equations, we find:
b = -2 or b = -3



Finding the Values of a and c

Now that we have the values of b, we can substitute them back into the first equation to find the corresponding values of a.

For b = -2:
a + (-2) = -5
a - 2 = -5
a = -3

For b = -3:
a + (-3) = -5
a - 3 = -5
a = -2

Therefore, we have two possible quadratic polynomials:
f(x) = -3x^2 - 2x + 6
f(x) = -2x^2 - 3x + 6

Conclusion

The quadratic polynomial whose sum and product of zeroes are -5 and 6 can be represented by two equations:
f(x) = -3x^2 - 2x + 6
f(x) = -2x^2 - 3x + 6