Question

6x² -3-7x find the zeros of the following quadratic polynominals and verify the relationship between the zeros and the coefficients​

Answer 1

S O L U T I O N :

We have quadratic polynomial p(x) = 6x² - 3 - 7x & zero of the polynomial p(x) = 0

Using By Factorisation Method :

➛ 6x² - 7x - 3 = 0

➛ 6x² + 2x - 9x - 3 = 0

➛ 2x(3x + 1) - 3(3x + 1) = 0

➛ (3x + 1) (2x - 3) = 0

➛ 3x + 1 = 0 Or 2x - 3 = 0

➛ 3x = -1 Or 2x = 3

➛ x = -1/3 Or x = 3/2

∴ α = -1/3 & β = 3/2 are two zeroes of the given polynomial.

As we know that quadratic polynomial compared with ax² + bx + c;

• a = 6
• b = -7
• c = -3

Now,

Sum of the zeroes :

➛ α + β = -b/a = (coefficient of x/coefficient of x²)

➛ -1/3 + 3/2 = -(-7)/6

➛ -1/3 + 3/2 = 7/6

➛ -2+9/6 = 7/6

➛ 7/6 = 7/6

Product of the zeroes :

➛ α × β = c/a = (constant term/coefficient of x²)

➛ -1/3 × 3/2 = -3/6

➛ -3/6 = -3/6

Thus,

The relationship between zeroes and coefficient is verified .

Answer 2

Given:

• Quadratic Polynomial p(x) = 6x² -3 -7x.

• Zero of the Quadratic Polynomial p(x) = 0.

To Find:

• Zeroes of the Quadratic Polynomial and verify the relationship between the Zeroes & it's Coefficients.

Solution:

$\textit{1)}$

We can find out the Zeroes of the Quadratic Polynomial with the help of Factorization Method.

Applying Factorization Method :

$\\ \longrightarrow\sf{6 {x}^{2} - 7x - 3 = 0} \\ \\ \longrightarrow\sf{6 {x}^{2} + 2x - 9x - 3 = 0} \\ \\ \longrightarrow\sf{2x(3x + 1) - 3(3x + 1) = 0} \\ \\ \longrightarrow\sf{(3x + 1)(2x - 3) = 0} \\ \\ \longrightarrow\sf{3x + 1 = 0 \: or \: 2x - 3 = 0} \\ \\ \longrightarrow\sf{3x = 0 - 1 \: or \: 2x = 0 + 3} \\ \\ \longrightarrow\sf{3x = - 1 \: or \: 2x = 3} \\ \\ \longrightarrow{\underline{\boxed{\sf{x = - \frac{1}{3} \: x = \frac{3}{2} }}}} \: \star$

Thus,

$\therefore$ The two zeroes of the Quadratic Polynomial are -1/3 and 3/2.

$\textit{2)}$

Finding the Sum of zeroes :

$\\ \longrightarrow\sf{ \alpha + \beta = - \frac{b}{a} } \\ \\ \longrightarrow\sf{ - \frac{1}{3} + \frac{3}{2} = - \frac{ - 7}{6} } \\ \\ \longrightarrow\sf{ - \frac{1}{3} + \frac{3}{2} = \frac{7}{6} } \\ \\ \longrightarrow\sf{ \frac{ - 2 + 9}{6} = \frac{7}{6} } \\ \\ \longrightarrow{\underline{\boxed{\sf{ \frac{7}{6} = \frac{7}{6} }}}} \: \star$

Thus,

$\therefore$ Relationship between the Zeroes of Quadratic Polynomial are verified.

$\textit{3)}$

Finding Product of Zeroes :

$\\ \longrightarrow\sf{ \alpha \times \beta = \frac{c}{a} } \\ \\ \longrightarrow\sf{ - \frac{1}{3} \times \frac{3}{2} = - \frac{3}{6} } \\ \\ \longrightarrow{\underline{\boxed{\sf{ - \frac{3}{6} = - \frac{3}{6} }}}} \: \star$

Thus,

$\therefore$ Relationship between the Product of Zeroes of Quadratic Polynomial are verified.