Question

find the zeroes of the quadratic polynomial and verify the relationship between the zeroes and the coefficients
$x {}^{2} - x - 72$
​

Answer 1

GIVEN

$p(x) = {x}^{2} - x - 72$

TO FIND :- 1. Zeros of p(x)

2.Verify the relation of zeros.

SOLUTION

We can find zeros of the polynomial by equating the value with zero.

${x}^{2} - x - 72 = 0 \\ \\ \\ \implies {x}^{2} - 9x + 8x - 72 = 0 \\ \\ \\ \implies x( x - 9) + 8(x - 9) = 0 \\ \\ \\ \implies (x - 9)(x + 8) = 0 \\ \\ \\ \implies \boxed{x = 9 \: or \: x = - 8}$

Let alpha and beta be the zeros of the polynomial.

Now we are in position to verify the following relation of coefficients and roots of a polynomial.

sum of zeros = - b/a

Product of zeros = c/a

where, a = coefficient of x^2

b = coefficient of x

c = constant term

$\alpha + \beta \\ \\ = - 8 + 9 \\ \\ = 1 \\ \\ = \frac{ 1}{1} \\ \\ = \frac{ - b}{a} \: \: (proved)$

$\alpha \beta \\ \\ = - 9 \times 8 \\ \\ = - 72 \\ \\ = \frac{ - 72}{1} \\ \\ = \frac{c}{a} \: \: (proved)$

Hence we have verified the relation.

Answer 2

$\large{\bold{ \fbox{ \fbox{ \bold{Solution :- \: }}}}}$

$\to \bold{ x {}^{2} - x - 72 = 0 } \\ \to \bold{{x}^{2} - 9x + 8x - 72 = 0 } \\ \to \: \bold{ x(x - 9) + 8(x - 9) = 0 } \\ \to \bold{ (x + 8)(x - 9) = 0 } \\ \to \bold{ x = - 8 } \: \: \bold{ or } \: \: \bold{ x = 9 }$

$\large{\fbox{ \fbox{ \bold{Verification :- \: }}}}$

$\large{ \to} \: \fbox{ \fbox{\bold{ \bold{ \alpha + \beta }= \bold{\frac{ - b}{a} }}}} \\ \to \bold{- 8 + 9 \: = \frac{ - ( - 1)}{1}} \\ \to \bold{ 1 = 1}$

$\large{ \to} \: \bold{ \fbox{ \fbox{ \bold{ \alpha \times \beta = \bold{\frac{c}{a} }}}}} \\ \to \bold{ \: - 8 \times 9 = \frac{ - 72}{1} } \\ \to \bold{- 72 = - 72}$

$\underline{\bold{\large{ \: \: Hence \: Verified \: \: \: \: }}}$