Question

find the zeroes of polynomial 4x² - 3 - 4x. also verify relation between zeroes and it's coefficient​

Answer 1

Step-by-step explanation:

Given:-

• A quadratic polynomial = 4x² - 3 - 4x

To Find:-

• The zeroes of the polynomial.

Solution:-

$\sf4 {x}^{2} - 3 - 4x$

$\sf4 {x}^{2} - 4x - 3$

By splitting the middle term:-

$\sf4 {x}^{2} - 6x + 2x - 3$

$\sf = 2x(2x- 3) + 1(2x - 3)$

$\sf = (2x + 1) (2x - 3)$

To find the zeroes, f(x) = 0

$\sf \dashrightarrow(2x + 1) (2x - 3) = 0$

$\sf \dashrightarrow2x + 1 = 0 \: \: \: (or) \: \: \: \: 2x - 3 = 0$

$\sf \dashrightarrow x= \dfrac{ - 1}{2} \: \: \: (or) \: \: \: \: x = \dfrac{3}{2}$

$\sf \therefore \underline{\boxed{\sf The \: zeroes \: of \: the \: polynomial \: are \: \dfrac{-1}{2} \: and \: \dfrac{3}{2}}}$

Now, Let us verify the relation between the zeroes and coefficients of the polynomial.

$\sf Sum \: of \: zeroes = \dfrac{ - 1}{2} + \dfrac{3}{2} = \dfrac{2}{2} = 1$

$\sf = \dfrac{ - b}{a} = \dfrac{ - ( - 4)}{4} = 1 = \boxed{ \sf \dfrac{ - (Coefficient \: of \: x)}{Coefficient \: of \: {x}^{2} }}$

$\sf Product \: of \: zeroes = \dfrac{ - 1}{2} \times \dfrac{3}{2} = \dfrac{-3}{4}$

$\sf = \dfrac{ c}{a} = \dfrac{ -3}{4} = \boxed{ \sf \dfrac{ Constant \: term}{Coefficient \: of \: {x}^{2} }}$

Answer 2

Answer :-

• Zeroes of the polynomial are - 1/2 and 3/2 respectively.

Given :-

• A quadratic polynomial 4x² - 3 - 4x.

To Find :-

• Zeroes of the polynomial.

Solution :-

Here's a quadratic polynomial 4x² - 3 - 4x

We can also write this as 4x² - 4x - 3

Let's find it's zeroes

4x² - 4x - 3

⇒ 4x² - 6x + 2x - 3

⇒ 2x (2x - 3) + 1 (2x - 3)

⇒ (2x + 1) (2x - 3)

____________________

• 2x + 1 = 0

⇒ 2x = 0 - 1

⇒ 2x = - 1

⇒ x = - 1/2

• 2x - 3 = 0

⇒ 2x = 0 + 3

⇒ 2x = 3

⇒ x = 3/2

____________________

Now let's verify the relation between it's zeroes and coefficients.

• Sum of zeroes => - 1/2 + 3/2 => 2/2 => 1

→ - b/a => - (-4)/4 => 1 => - (coefficient of x)/coefficient of x²

• Product of zeroes => - 1/2 × 3/2 => - 3/4

→ c/a => - 3/4 => constant term/coefficient of x²

Hence, Verified !