Question

find the zero of each of the following quadratic polynomials and verify the relationship between the zeros and the coefficient of X square + 7 x + 12​

Answer 1

$\bold {\huge{Solution-:}}$

$\bold {\underline {Answer}}$

$\bold {Polynomial}$ = ${x}^{2}$ + 7x + 12

By splitting the middle term we can find the zeros of the given polynomial

${x}^{2}$ + 7x + 12

${x}^{2}$ + 4x + 3x + 12

x (x + 4) + 3 (x + 4)

(x + 3) (x + 4)

These are the factors of the polynomial

$\bold {To\ find\ the\ zeros -}$

x + 3 = 0

x = -3

x + 4 = 0

x = -4

$\bold {So\ (-3)\ and\ (-4)\ are\ the\ two\ zeros }$

Now,

$\bold {\underline{Relationship\ between\ zeros\ and\ the\ coefficient.}}$

Here the polynomial is given in the form a ${x}^{2}$ + bx + c

Coefficient of ${x}^{2}$= 1

$\bold {Coefficient\ of\ x\ =\ 7}$

$\bold {Constant\ term\ =\ 12}$

$\bold {Sum\ of\ its\ zeros\ =\ (-3)\ +\ (-4)\ =\ -7}$

$\boxed{\bold {\dfrac {-b}{a} = \dfrac{-7}{1} = \dfrac{Coefficient\ of\ x}{Coefficient\ of\ x^2}}}$

$\bold {Product\ of\ its\ zeros\ =\ (-3)\ ×\ (-4)\ =\ 12}$

$\boxed{\bold{\dfrac {c}{a} = \dfrac{12}{1} =\dfrac{Constant\ term}{Coefficient\ of\ x^2}}}$

$\bold {\underline {Hence\ verified}}$