Question

if alpha and beta are the zeros of the quadratic polynomial x2+7x+3 then the value of (alpha-beta)^2
​

Answer 1

Step-by-step explanation:

alpha + beta = -7

alpha* beta = 3

(alpha - beta)²

= (alpha + beta)² - 4alpha*beta

= (-7)² - 4(3)

= 49 - 12

= 37

Answer 2

Solution :

$\bf{\underline{\bf{Given\::}}}}}$

If α and β are the zeroes of the quadratic polynomial x² + 7x + 3

$\bf{\underline{\bf{To\:find\::}}}}}$

The value of (α - β)²

$\bf{\underline{\bf{Explanation\::}}}}}$

As we know that polynomial are compared with ax² + bx + c

• a = 1
• b = 7
• c = 3

$\bf{\underline{1_{st}\:Condition\::}}$

We now that sum of the zeroes;

$\longrightarrow\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } }\\\\\\\longrightarrow\sf{\alpha +\beta =\dfrac{-7}{1} }\\\\\\\longrightarrow\bf{\alpha +\beta =-7}}$

$\bf{\underline{2_{nd}\:Condition\::}}$

We know that product of the zeroes;

$\longrightarrow\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:x^{2} } }\\\\\\\longrightarrow\sf{\alpha \times \beta =\dfrac{3}{1} }\\\\\\\longrightarrow\bf{\alpha \times \beta =3}}$

Now;

We know that formula of the (α - β)² :

$\implies\sf{(\alpha +\beta )^{2} -4\alpha \beta }\\\\\implies\sf{(-7)^{2} -4(3)}\\\\\implies\sf{-7\times (-7)-4\times 3}\\\\\implies\sf{49-12}\\\\\implies\bf{\red{37}}$

Thus;

The value of (α - β)² = 37 .