Question

find the zeros of quadratic polynomial P(x)=x²-x-72 and find the relationship between zeros and its coefficient ​

Answer 1

AnswEr:-

Given:-

• P(x)=x²-x-72

To find:-

• Zeroes of the polynomial & the relationship btw zeroes and co-efficients.

Solution:-

Given , P(x) = x²-x-72

⇒ x²-x-72=0

⇒ x²-9x+8x -72 =0

⇒ x(x-9) +8(x-9)  =0

⇒ (x+8)(x-9)=0

⇒ x+8 = 0   (OR)  x-9 = 0

⇒ x = -8 , 9

∴ Zeroes of polynomial are -8 ,9.

Relationship btw zeroes and co-efficients:-

If ax²+bx+c = 0 is a polynomial , where a≠0 & αβ are zeroes of the polynomial ,

❥ αβ = c/a

❥ α+β = -b/a

According to question :

➾ αβ = c/a

⇒ (9)(-8) = -72/1

⇒ - 72 = -72

⇒ 1 = 1 .

➾ α + β = -b/a

⇒ 9 + (-8) = -(-1)/1

⇒ 1 = 1 .

Hence verified!

____________________

Answer 2

$\sf \underline{Answer:-}$

$\tt \: Let \: \: p(x) = {x}^{2} - x - 72$

$\tt \: Zero \: \: of \: \: the \: \: polynomial \: \: is \: \: the \: \: value \: \: of \: \: x \: \: where \: \: p(x) = 0$

$\tt \: Putting \: p(x) = 0$

$\bf \: {x}^{2} - x - 72 = 0$

$\tt \: By \: \: splitting \: \: the \: \: middle \: \: term \:$

$\bf \: {x}^{2} - 9x + 8x - 72 = 0$

$\bf \: x(x - 9) + 8(x - 9) = 0$

$\bf \: (x + 8)(x - 9) = 0$

$\tt \: So , \: x = - 8 \: ,9$

Therefore,

$\bf \: \alpha = - 2 \: \: and \: \: \beta = - 5 \: \: are \: \: the \: \: zeroes \: \: of \: \: the \: \: polynomial \:$

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

$\bf \: Comparing \: \: with \: \: {ax}^{2} + bx + c \:$

So,

• a = 1
• b = -1
• c = -72

we have to verify,

$\bf \: Sum \: \: of \: \: zeroes \: \: = \: \frac{coefficient \: \: of \: \: x \: }{coefficient \: \: of \: \: {x}^{2} }$

$\sf \: i.e. \: \alpha + \beta = \frac{ - b}{a}$

$\bf \: LHS \: = \alpha + \beta$

$\bf \: = ( - 8) + 9$

$\bf \: = 1$

$\bf \: RHS = \frac{ - b}{a}$

$\bf \: = \frac{ - ( - 1)}{1}$

$\bf \: = 1$

$LHS = RHS$

now,

$\bf \: Product \: \: of \: \: zeroes \: \: = \: \frac{constant \: \: term}{coefficient \: \: of \: \: {x}^{2} }$

$\sf \: i.e. \: \alpha \beta = \frac{c}{a}$

$\bf \: LHS \: = \alpha \beta$

$\bf \: = ( - 8)(9)$

$\bf \: = - 72$

$\bf \: RHS = \frac{c}{a}$

$\bf \: = \frac{ - 72}{1}$

$\bf \: = - 72$

$LHS = RHS$

$\bf {\fbox{ \red{ </strong><strong>♡</strong><strong>Hence \: \: </strong><strong>Verified</strong><strong> </strong><strong>♡</strong><strong>}}}$