Question

10. If a and B are zeroes of a quadratic polynomial P(x) = 4x2-5x-1. Find the value of a²?ß+aß²​

Answer 1

Answer :-

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

Here the concept of Quadratic Polynomials has been used. If we are given a Quadratic Polynomial of form p(x) = ax² + bx + c then its zeroes will be α and β. Here Quadratic Polynomial intersects x - axis twice and gives two solutions of the same polynomial.

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★ Formula Used :-

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

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

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★ To Find :-

$\: \\ \: \large{\rm{\alpha^{2} \beta \: \: + \: \: \beta^{2} \alpha \: \: = \: \: \bf{?}}}$

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★ Question :-

If α and β are zeroes of a quadratic polynomial p(x) = 4x² - 5x - 1. Find the value of αβ + β²α

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★ Solution :-

Given,

p(x) = 4x² - 5x + 1

Here, a = 4 , b = -5 and c = 1

Let's solve it by easier method. Here we are not gonna find out roots of this solution, because its not asked. Easily we can solve it by Coefficient Rule.

Coefficient Rule :- It states that if we are given the ratios of two values, we cab simply apply its coefficient in the values and use it.

Then, according to the question :-

~ For the value of α + β :-

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

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: \alpha \: + \: \beta \: \: = \: \: \bf{\dfrac{-(-5)}{4}}}}$

$\: \\ \: \large{\boxed{\sf{\alpha \: + \: \beta \: \: = \: \: \bf{\dfrac{5}{4}}}}}$

~ For the value of αβ :-

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: \alpha \: \times \: \beta \: \: = \: \: \bf{\dfrac{(c)}{a}}}}$

$\: \\ \qquad \: \large{\sf{\Longrightarrow \: \: \: \alpha \: \times \: \beta \: \: = \: \: \bf{\dfrac{(1)}{4}}}}$

$\: \\ \: \large{\boxed{\sf{\alpha \: \times \: \beta \: \: = \: \: \bf{\dfrac{1}{4}}}}}$

~ For the value of α²β + β²α :-

$\: \\ \qquad \: \large{\rm{\Longrightarrow \: \: \alpha^{2} \beta \: \: + \: \: \beta^{2} \alpha \: \: = \: \: \bf{\alpha \beta(\alpha \: + \: \beta)}}}$

Here we have just shortened and simplified the equation.

Now we already have the value of αβ and α + β.

Then, applying those, we get

$\: \\ \qquad \: \large{\rm{\Longrightarrow \: \: \: \alpha^{2} \beta \: \: + \: \: \beta^{2} \alpha \: \: = \: \: \bf{\dfrac{1}{4} \: \times \dfrac{5}{4} \: \: = \: \: \underline{\underline{\dfrac{5}{16}}}}}}$

$\: \\ \: \large{\boxed{\sf{\alpha^{2} \beta \: \: + \: \: \beta^{2} \alpha\: \: = \: \: \bf{\dfrac{5}{16}}}}}$

$\: \\ \large{\underline{\underline{\sf{\leadsto \: \: Hence, \: the \: value \: of \: \alpha^{2} \beta \: \: + \: \: \beta^{2} \alpha \: \: is \: \: \: \boxed{\bf{\dfrac{5}{16}}}}}}}$

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$\: \: \qquad \large{\underline{\underline{\rm{\mapsto \: \: \: Let's \: know \: more \: :-}}}}$

• Polynomials are the statements written which shows that the value of x is rhe solution of that statement.

• Quadratic Equations are the equations formed using Variable and Constant terms but the degree variable term shows the number of zeroes of it. For quadratic equation that degree is 2.

• Linear Equation is the equation formed using constant and variable terms but these terms are of single degrees.

* Note :- Here I have solved this using direct application method. You can solve this question by finding out the zeroes also. For this type of answer, please refer to answer given by @itzVanquisher.

Answer 2

Correct question :

10. If a and B are zeroes of a quadratic polynomial P(x) = 4x² - 5x + 1 . Find the value of a²ß+aß²​

Given :

10. If α and β are zeroes of a quadratic polynomial P(x) = 4x² - 5x + 1.

To find :

Find the value of α²β + αβ²

Solution :

Given polynomial :

⇒ P(x) = 4x² - 5x + 1

⇒ 4x² - 5x + 1 = 0

⇒ 4x² - 4x - x + 1 = 0

⇒ 4x(x - 1) - 1(x - 1) = 0

⇒ (4x - 1) (x - 1) = 0

⇒ 4x = 1  or,  x = 1

⇒ x = 1/4  or,  x = 1

∴ α = 1/4 and β = 1

Now putting values,

⇒ α²β + αβ²

⇒ (1/4)² * 1 + 1/4 * 1²

⇒ 1/16 + 1/4

⇒ (1 + 4)/16

⇒ 5/16

Therefore,

Value of α²β + αβ² = 5/16