Question

if alpha and beta are the zeros of the given polynomial p(x) = 4x^2 - x - 4.find 1/2alpha + 1/2beta
(class 10th maths chapter polynomial) ​

Answer 1

Answer:-

Given:

α & β are the zeroes of 4x² - x - 4.

On comparing with the standard form of a quadratic equation i.e., ax² + bx + c = 0 ,

let ,

• a = 4
• b = - 1
• c = - 4

We know that,

Sum of the zeroes = - b/a

⟶ α + β = - ( - 1) / 4

⟶ α + β = 1/4

Multiply 1/2 both sides.

⟶ 1/2 (α + β) = (1/2)(1/4)

⟶ 1/2(α) + 1/2(β) = 1/8

∴ The value of 1/2(α) + 1/2(β) is 1/8.

Answer 2

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Zeroes of Quadratic polynomials have been used. If a quadratic polynomial ax² + bx + c = 0 is given, then its zeroes will be α and β. And here the coefficients are given as :-

a = coefficient of x², b = coefficient of x and c = constant.

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★ Formula Used :-

$\: \: \large{\boxed{\boxed{\bf{\alpha \: + \: \beta \: = \: \dfrac{(-b)}{a}}}}}$

$\: \: \large{\boxed{\boxed{\bf{\alpha \: \times \: \beta \: = \: \dfrac{c}{a}}}}}$

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★ Question :-

If alpha and beta are the zeros of the given polynomial p(x) = 4x² - x - 4. Find 1/2alpha + 1/2beta.

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★ To find :-

$\: \large{\tt{\dfrac{1}{2} \alpha \: + \: \dfrac{1}{2} \beta \: = \: ?}}$

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★ Solution :-

Given,

➺ p(x) = 4x² - x - 4

Here let the zeroes of p(x) be α and β.

Here, a = 4 , b = (-1) and c = (-4).

Then, according to the formula, we get,

$\\ \large{\bf{\longmapsto \: \: \: \alpha \: + \: \beta \: = \: \dfrac{(-b)}{a}}}$

Dividing all the terms by 2, we get,

$\\ \large{\bf{\longmapsto \: \: \dfrac{\alpha}{2} \: + \: \dfrac{\beta}{2} \: = \: \dfrac{\dfrac{(-b)}{a}}{2}}}$

$\: \\ \large{\bf{\longmapsto \: \: \dfrac{1}{2} \alpha \: + \: \dfrac{1}{2} \beta \: = \: \dfrac{(-b)}{2a}}}$

By applying the values, we get,

$\: \\ \large{\bf{\longmapsto \: \: \: \dfrac{1}{2} \alpha \: + \: \dfrac{1}{2} \beta \: = \: \dfrac{-(-1)}{2 \: \times \:4}}}$

$\: \\ \large{\boxed{\bf{\dfrac{1}{2} \alpha \: + \: \dfrac{1}{2} \beta \: = \: \dfrac{1}{8}}}}$

$\\ \large{\overbrace{\underbrace{\sf{Thus \: the \: required \: answer \: is \: \dfrac{1}{8}}}}}$

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$\: \: \underline{\underline{\rm{\Longrightarrow \: \: Confused? \: Don't \: worry \: let's \: verify \: it \: :-}}}$

For verification we need to simply apply the values we got. Then,

➣ α + β = -b/a

➣ ½α + ½β = ¼ × ½

➣ ½(α + β) = ¼ × ½

➣ ½(α + β) = ⅛

Clearly, we got the desired answer. So our answer is correct.

Hence, verified.

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$\: \: \: \huge{\boxed{\tt{\large{More \: to \: know \: :-}}}}$

• Polynomials are the equations formed using constant and linear terms but can be of many degrees.

• Linear Equations are the equations formed using constant and linear terms but of single degrees.

• Different types of Polynomials :-

• Linear Polynomials
• Quadratic Polynomials
• Cubic Polynomials
• Bi - Quadratic Polynomials