if one zero of f(x)=4x²-8kx+8x-9 is negative of the other, them find the zeros of kx²+kx+2
Answers
Step-by-step explanation:
The zeroes are -1 and -2
Step-by-step explanation:
\text{Given that one zero of quadratic polynomial }4x^2-8kx+8x-9 \text{ is negative of other then}Given that one zero of quadratic polynomial 4x
2
−8kx+8x−9 is negative of other then
\text{we have to find the zeroes of }kx^2+3kx+2we have to find the zeroes of kx
2
+3kx+2
\text{ As one zero of quadratic polynomial }4x^2-8kx+8x-9 \text{ is negative of other } As one zero of quadratic polynomial 4x
2
−8kx+8x−9 is negative of other
∴ let the zeroes are α and -α
\text{sum of roots= }\alpha+(-\alpha)=\frac{-b}{a}=\frac{-(-8k+8)}{4}sum of roots= α+(−α)=
a
−b
=
4
−(−8k+8)
⇒ \frac{-(-8k+8)}{4}=0
4
−(−8k+8)
=0
⇒ \frac{8k-8}{4}=0
4
8k−8
=0
8k-8=08k−8=0
k=1k=1
The polynomial becomes
(1)x^2+3(1)x+2(1)x
2
+3(1)x+2
x^2+3x+2x
2
+3x+2
x^2+2x+x+2x
2
+2x+x+2
x(x+2)+1(x+2)x(x+2)+1(x+2)
(x+1)(x+2)(x+1)(x+2)
x+1=0 ⇒ x=-1
x+2=0 ⇒ x=-2
Hence, the zeroes are -1 and -2
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Answer:
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 ) =α+(−α)= coefficient of x / coefficient of x²
\alpha - \alpha = \frac{-8k+8}{4}α−α=
4
−8k+8
→ ( As per to given )
0 = -2k + 20=−2k+2 → ( After simplifying )
2k-22k−2
k = \frac{2}{2} = 1k=
2
2
=1
Hence the value of k is 1
----------------------------------------------------------------------------------------------
⇒ 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 + 21x
2
+3x+2
→ Now , we can find the zeroes of polynomial by spliting the middle term method for 1 x^{2} + 3 x + 21x
2
+3x+2 -
x^{2} + 3x + x = 0x
2
+3x+x=0
x^{2} + 2x+ x+ 2 = 0x
2
+2x+x+2=0
x ( x + 2 ) + 1 ( x + 2 ) = 0x(x+2)+1(x+2)=0
(x + 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 = 0x+1=0
x = - 1x=−1
→Alpha - α = -1
----------------------------------------------------------------------------------------------
Another Zero :-
x + 2 = 0x+2=0
x = -2x=−2
→Beta - β = -2
Hence the two zeroes are -1 and -2.