Question

find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficient today's x2 - X - 72​

Answer 1

Answer :

The zeroes of the given polynomial are -8 and 9

Given :

The quadratic polynomial is :

• x² - x - 72

Task :

• To find the zeroes of the given polynomial
• To verify the relationship between the zeroes

Solution :

$\sf x^{2}-x-72 \\\\ \sf = x^{2} - 9x + 8x - 72 \\\\ \sf = x(x - 9) + 8(x - 9) \\\\ \sf = (x + 8)(x-9)$

Thus the zeroes are :

$\sf x + 8 = 0 \: \: and \: \: x - 9 = 0 \\\\ \sf \implies x = -8 \: \: and \: \implies x = 9$

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Verification of the relationships :

$\sf Sum \: of \: the \: zeroes = -\dfrac{Coefficient \: of \: x}{Coefficient \: of \: x^{2}} \\\\ \sf \implies -8 + 9 = -\dfrac{-1}{1}\\\\ \sf \implies 1 = 1$

$\sf Product \: of \: the \: zeroes=\dfrac{Constant \: term}{Coefficient \: of \: x^{2}} \\\\ \sf \implies (-8)\times 9 = -72 \\\\ \sf \implies -72 = -72$

$\bf Hence \: \: verified$

Answer 2

Step-by-step explanation:

GIVEN:

X^2 -X-72.

TO FIND:

ZEROES OF THIS QUADRATIC POLYNOMIAL.

METHOD:

MIDDLE TERM FACTORISATION.

SOLUTION:

${x}^{2} - x - 72 \\ \\ = {x}^{2} + 8x - 9x - 72 \\ \\ = x(x + 8) - 9(x + 8) \\ \\ = (x - 9)(x + 8)$

NOW THE ZEROES ARE:

X-9=0=>X=9

OR,

X+8=0=>X=-8

VERIFICATION OF RELATIONSHIPS:

SUM OF ZEROES = COEFFICIENT OF X/COEFFICIENT OF X^2

=>-8+9=-1/1

=-1=-1 (VERIFIED)

PRODUCT OF ZEROES = CONSTANT TERM/COEFFICIENT OF X^2

=>-8×9=-72/1

= -72=-72 (VERIFIED)