Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient 3x² 5x - 2?

Question

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient 3x²   5x - 2?

Answer

Zeroes of Quadratic Polynomial

The quadratic polynomial is given as 3x² + 5x - 2.

To find the zeroes of the polynomial, we need to solve for x such that the polynomial is equal to zero.

Thus, we have the equation 3x² + 5x - 2 = 0.

We can solve this equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 3, b = 5, and c = -2.

Substituting these values in the formula, we get:

x = (-5 ± √(5² - 4(3)(-2))) / 2(3)

Simplifying, we get:

x = (-5 ± √49) / 6

Thus, the zeroes of the quadratic polynomial are:

x = (-5 + 7) / 6 = 1/3

x = (-5 - 7) / 6 = -2

Relationship between Zeroes and Coefficient

Let the zeroes of the quadratic polynomial be α and β.

Then, the polynomial can be written as:

3x² + 5x - 2 = 3(x - α)(x - β)

Expanding the right-hand side, we get:

3x² - 3(α + β)x + 3αβ

Comparing the coefficients of the quadratic term, linear term, and constant term, we get:

α + β = -5/3

αβ = -2/3

Now, substituting the values of α and β, we get:

α + β = 1/3 - 2 = -5/3

αβ = (1/3)(-2) = -2/3

Thus, we have verified the relationship between the zeroes and the coefficient of the quadratic polynomial.

Answer

3×²+5×-2
3x²+2x-3×-2
x(3x +2 ) -1 ( 3x+2)
(x-1) (3x+2)
x=1 × = -2/3
alpha = 1 beta = -2/3

alpha + beta = -b/a
1 + -2/3 = -(+5/3)
-5/3 =-5/3

alpha × beta = c/ a
1× -2/3 = -2/3
-2/3 = -2/3

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