Question

if one zero of the quadratic polynomial f(x)=4x^2-8kx+8x-9 is negative of the other ,then find the zeros of kx^2+3kx+2.

Answer 1

According to the given question it is already given that - ''If one zero of the quadratic polynomial f(x)=4 x² - 8 k x + 8 x - 9  is negative''

So let the roots of this polynomial be as follows :-

⇒ α , - α { As per to given )

We already know what is the sums of roots , ( i.e ...)

$\alpha + ( - \alpha ) =$ coefficient of x / coefficient of x²

$\alpha - \alpha = \frac{-8k+8}{4}$    → ( As per to given )

$0 = -2k + 2$                        → ( After simplifying )

$2k-2$

$k = \frac{2}{2} = 1$

Hence the value of k is 1

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⇒ Now let us put the value of k as 1 in the given polynomial - kx² + 3 kx + 2

⇒ So we get as follows ,

⇒$1 x^{2} + 3 x + 2$

→ Now , we can find the zeroes of polynomial by spliting  the middle term method for $1 x^{2} + 3 x + 2$ -


$x^{2} + 3x + x = 0$

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

$x ( x + 2 ) + 1 ( x + 2 ) = 0$

$(x + 1 ) ( x+ 2) = 0$ 

→ Now , let us find the alpha and beta for the following polynomial to get the final answer :-

$x + 1 = 0$

$x = - 1$


→Alpha - α = -1 

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Another Zero :-

$x + 2 = 0$

$x = -2$


→Beta - β =  -2

Hence the two zeroes are -1 and  -2.

Answer 2

Answer:

The roots of the equation are  -2 and -1

Step-by-step explanation:

Given,

One zero of the quadratic polynomial 4x² - 8kx +8x -9 is negative of the other

To find,

The zeros of kx² +3kx +2

Recall the concepts:

1. For a quadratic equation ax ²+bx +c = 0 Sum of roots = $\frac{-b}{a}$ and Product of roots = $\frac{c}{a}$

Solution:

Let 'α' be one root of the  4x² - 8kx +8x -9, then given that the other root is '-α'

4x² - 8kx +8x -9 =  4x² - 8(k-1)x -9 = 0

Comparing the above equation  with ax ²+bx +c = 0 we get

a = 4, b= -8(k-1) and c= -9

From the formula (1),  for the equation 4x² - 8(k-1)x -9 = 0  

Sum of roots = $\frac{-b}{a}$ = $\frac{8(k-1)}{4}$ = 2(k-1)

Since 'α' and '-α' are roots, sum of roots =  α + -α = 0

2(k-1) = 0

k-1 = 0

k =1

Substituting the value of k in the equation  kx² +3kx +2, the equation becomes

kx² +3kx +2 = x² +3x +2

Required to find the roots of the equation x² +3x +2 = 0

To find the roots of the equation x² +3x +2 = 0

To find the roots of the equation x² +3x +2 by splitting the middle term, we need to find two numbers such that their sum = 3 and product = 2

Two such numbers are 2 and 1

Then,

x² +3x +2 = x² +2x +x +2

= x(x+2)+1(x+2)

= (x+2)(x+1)

x² +3x +2  = 0  ⇒(x+2)(x+1) = 0

⇒(x+2) = 0, (x+1) = 0

⇒x= -2 , x = -1

Hence the roots of the equation are  -2 and -1

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