Question

. Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.
x2 - 2x - 8
​

Answer 1

Solution :

We have quadratic polynomial p(x) = x² - 2x - 8;

zero of the polynomial p(x) = 0

$\longrightarrow\sf{x^{2} -2x-8=0}\\\\\longrightarrow\sf{x^{2} +2x - 4x-8=0}\\\\\longrightarrow\sf{x(x+2) - 4(x+2) = 0}\\\\\longrightarrow\sf{(x+2)(x-4)=0}\\\\\longrightarrow\sf{x+2=0\:\:\:Or\:\:\:x-4=0}\\\\\longrightarrow\bf{x=-2\:\:\:Or\:\:\:x=4}$

∴ α = -2 & β = 4 are the zeroes of the polynomial .

As we know that given quadratic polynomial compared with ax² + bx + c;

• a = 1
• b = -2
• c = -8

Now;

$\underline{\mathcal{SUM\:OF\:THE\:ZEROES\::}}$

$\mapsto\tt{\alpha +\beta =\dfrac{-b}{a} =\bigg\lgroup\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} }\bigg\rgroup}\\\\\\\mapsto\tt{-2+4=\dfrac{-(-2)}{1} }\\\\\\\mapsto\bf{2=2}$

$\underline{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}$

$\mapsto\tt{\alpha \times \beta =\dfrac{c}{a} =\bigg\lgroup\dfrac{Constant\:term}{Coefficient\:of\:x^{2} }\bigg\rgroup}\\\\\\\mapsto\tt{-2\times 4=\dfrac{-8}{1} }\\\\\\\mapsto\bf{-8=-8}$

Thus;

The relationship between zeroes & coefficient is verified .