Question

find the zeroes of the quadratic polynomial x2+7x+10 and verify the relationship between the zeroes and the cofficient.​

Answer 1

Solution :

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

The quadratic polynomial x² + 7x + 10

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

The zeroes and verify the relationship between the zeroes and the coefficient.

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

We have p(x) = x² + 7x + 10

Zero of the polynomial is p(x) = 0

So;

$\longrightarrow\sf{x^{2} +7x+10=0}\\\\\longrightarrow\sf{x^{2} +2x+5x+10=0}\\\\\longrightarrow\sf{x(x+2)+5(x+2)=0}\\\\\longrightarrow\sf{(x+2)(x+5)=0}\\\\\longrightarrow\sf{x+2=0\:\:\:Or\:\:\:x+5=0}\\\\\longrightarrow\sf{\red{x=-2\:\:\:Or\:\:\:x=-5}}$

∴ The α = -2 and β = -5 are the zeroes of the polynomial.

As the given quadratic polynomial as we compared with ax²+bx+c=0

• a = 1
• b = 7
• c = 10

So;

$\blacksquare\bf{\green{\underline{\underline{\tt{Sum\:of\:the\:zeroes\::}}}}}$

$\mapsto\sf{\alpha +\beta=\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x^{2} }{Coefficient\:of\:x}} \\\\\\\mapsto\sf{-2+(-5)=\dfrac{-7}{1} }\\\\\\\mapsto\sf{-2-5=-7}\\\\\\\mapsto\sf{\red{-7=-7}}$

$\blacksquare\bf{\green{\underline{\underline{\tt{Product\:of\:the\:zeroes\::}}}}}$

$\mapsto\sf{\alpha \times \beta=\dfrac{c}{a} =\dfrac{Constant\:term }{Coefficient\:of\:x}} \\\\\\\mapsto\sf{-2\times (-5)=\dfrac{10}{1} }\\\\\\\mapsto\sf{-(-10)=10}\\\\\\\mapsto\sf{\red{10=10}}$

Thus;

Relationship between zeroes and coefficient is verified .

Answer 2

Solution :-

→ x² + 7x + 10 = 0

→ x² + 2x + 5x + 10 = 0

→ x(x + 2) + 5(x + 2) = 0

→ (x + 2)(x + 5) = 0

Putting Both Equal to zero now,

→ x + 2 = 0

→ x = (-2).

&

→ x + 5 = 0

→ x = (-5) .

_____________________

Now, First Relation is :-

→ Sum of Zeros = - (coefficient of x) /(coefficient of x²)

Putting both values ,

→ (-2) + (-5) = -(7)/1

→ (-7) = (-7) ✪✪ Hence Verified. ✪✪

Second Relation :-

→ Product Of Zeros = Constant Term / (coefficient of x²)

Putting both Values ,

→ (-2) * (-5) = (10) / 1

→ 10 = 10 ✪✪ Hence Verified. ✪✪

______________________