Question

Verify the relationship between the zeroes and cofficent of the following polynomial Xsquare+11x+18

Answer 1

Question :-

Verify the relationship between the zeroes and coefficients of the following polynomial x²+ 11x + 18

Answer :-

Given :-

Quadratic equation :- x² + 11x + 18

Required to find :-

• Verify the relationship between the zeroes and coefficients

Solution :-

Given data :-

Quadratic equation :- x² + 11x + 18

we need to verify the relationship between the zeroes and the coefficients .

In order, to verify the relationship first let's find the zeroes of the polynomial .

So,

Let's Factorise the given Quadratic equation

x² + 11x + 18 = 0

sum = 9x + 2x = 11x

product = 9 x 2 = 18

x² + 9x + 2x + 18 = 0

x ( x + 9 ) + 2 ( x + 9 ) = 0

( x + 9 ) ( x + 2 ) = 0

Here,

( x + 9 ) ( x + 2 ) are the factors of p ( x )

Now,

Equal this factors with zero to find the values of zero ;

This implies ;

=> x + 9 = 0

=> x = - 9

=> α = - 9

=> x + 2 = 0

=> x = - 2

=> β = - 2

So,

The zeroes of the above quadratic equation is - 9 , - 2

Now,

Let's verify the relationship between the zeroes and the coefficients .

As we know that ;

$\boxed{ \tt{ \orange{ \bf \alpha + \beta = \frac{ - (coefficient \: of \: x)}{coefficient \: of \: {x}^{2} } }}}$

α + β =

=> - 9 + ( - 2 )

=> - 9 - 2

=> - 11

But,

$\tt{ \dfrac{ - (coefficient \: of \: x)}{coefficient \: of \: {x}^{2} }}$

=> - ( 11 )/1

=> - 11/1

=> - 11

Hence,

$\boxed{ \tt{ \bf \alpha + \beta = \frac{ - (coefficient \: of \: x)}{coefficient \: of \: {x}^{2} } }}$

Similarly,

$\boxed{ \tt{ \green{ \bf \alpha \beta = \frac{ constant \: term}{coefficient \: of \: {x}^{2} } }}}$

α β =

=> - 9 x - 2

=> 18

But,

$\tt{ \dfrac{constant \: term}{coefficient \: of \: {x}^{2} }}$

=> 18/1

=> 18

Hence,

$\boxed{ \tt{ \bf \alpha \beta = \frac{ constant \: term}{coefficient \: of \: {x}^{2} } }}$

Therefore,

The relationship between the zeroes and the coefficients had been verified .

Answer 2

Step-by-step explanation:

$\huge{\underline{\pink{QuesTion}}}$

Verify the relationship between the zeroes and cofficent of the following polynomial Xsquare+11x+18

TO FIND:

Verify the relationship between the zeroes and cofficent.

$\huge{\underline{\pink{AnsWer}}}$

Given

quadratic equation is x^2+11x+18

solution:

standard form of quadratic equation is

ax^2+bx+c=0

same as,

x²+11x+18

we need to find zeroes,

so,

ғᴀᴄᴛᴏʀɪᴇs ᴛʜᴇ ɢɪᴠᴇɴ ᴇǫᴜᴀᴛɪᴏɴ

${x}^{2} + 11x + 18 = 0$

${x}^{2} + 9x + 2x + 18 = 0$

$x(x + 9) + 2(x + 9)$

$(x + 2)(x + 9) = 0$

ɴᴏᴡ,

$x + 9 = 0$

$x = - 9(or) \alpha = - 9$

ᴀɴᴅ

$x + 2 = 0$

$x = - 2( \: or \: ) \beta = - 2$

ᴢᴇʀᴏᴇs ᴏғ ᴛʜᴇ ɢɪᴠᴇɴ ᴇǫᴜᴀᴛɪᴏɴ ɪs -9&-2

Verification:

$\alpha + \beta = - 9 + ( - 2) = - 9 - 2 = - 11$

we know that

$\alpha + \beta = \frac{ - b}{a}$

$\alpha + \beta = - 11$

same as,

$\alpha \beta = ( - 9) \times ( - 2) = 18$

$we \: know \: that$

$\alpha \beta = \frac{c}{a}$

$\alpha \beta = 18$

Hence verified.