Obtain a quadratic polynomial where sum of zeroes is 8/5 and product of zeroes is 3/5?

Question

Answer

Alpha+beta= 8/5=-b/a
Alpha* beta=3/5=c/a
therefore, a= 5
b=-8
c=3
hence the polynomial is 5x^2-8x+3

Answer

Answer

Quadratic Polynomial with Given Sum and Product of Zeroes

To obtain a quadratic polynomial where the sum of zeroes is 8/5 and the product of zeroes is 3/5, we can follow the steps below.

Step 1: Understand the Problem
We are given that the sum of the zeroes of the quadratic polynomial is 8/5 and the product of the zeroes is 3/5. We need to find the quadratic polynomial that satisfies these conditions.

Step 2: Identify the Quadratic Polynomial
Let's assume the quadratic polynomial is of the form:
$ax^2 + bx + c = 0$

Step 3: Use the Sum and Product of Zeroes Formulas
The sum of zeroes formula states that for a quadratic equation $ax^2 + bx + c = 0$, the sum of the zeroes is given by:
$-\frac{b}{a}$

The product of zeroes formula states that the product of the zeroes is given by:
$\frac{c}{a}$

Since we know the sum and product of the zeroes, we can set up the following equations:

Sum of zeroes: $-\frac{b}{a} = \frac{8}{5}$
Product of zeroes: $\frac{c}{a} = \frac{3}{5}$

Step 4: Solve the Equations
To solve the equations, we can use substitution. By rearranging the first equation, we have:
$b = -\frac{8}{5}a$

Substituting this value of b into the second equation, we get:
$\frac{c}{a} = \frac{3}{5}$

Multiplying both sides by a, we get:
$c = \frac{3}{5}a$

Step 5: Form the Quadratic Polynomial
Now that we have expressions for b and c in terms of a, we can substitute these values back into the quadratic polynomial form:
$ax^2 + bx + c = 0$

Substituting the values of b and c, we get:
$ax^2 - \frac{8}{5}ax + \frac{3}{5}a = 0$

Simplifying, we have:
$5ax^2 - 8ax + 3a = 0$

Therefore, the quadratic polynomial with the given sum and product of zeroes is:
$5ax^2 - 8ax + 3a = 0$

Summary:
To obtain a quadratic polynomial where the sum of zeroes is 8/5 and the product of zeroes is 3/5, we followed the steps of identifying the quadratic polynomial, using the sum and product of zeroes formulas, solving the equations, and forming the final quadratic polynomial. The resulting polynomial is $5ax^2 - 8ax + 3a = 0$.

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