If a and b are the zeros of the quadratic equation 2x² - 5x - 6 then form a quadratic polynomial whose zeros are a + b and ab
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P(x)=2x²-5x-6,having zeros as alpha and beta,
to find -->f(x) such that its zeros are (alpha +beta)and alpha×beta,
in p(x) sum of zeros=alpha+beta=-b/a,
alpha+beta=-(-5)/2=5/2,
product of zeros=alpha×beta=c/a,
alpha×beta=-6/2=-3,
zeros of f(x) are 5/2 and -3,
f(x)=x²-( sum of zeros)x+ product of zeros,
=x²-(5/2-3)x+(5/2×-3),
=x²-(-1/2)x-(15/2),
=x²+1/2x-15/2,
=2x²+x-15
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Step-by-step explanation:
ᴘ(x)=2x²-5x-6,ʜᴀᴠɪɴɢ ᴢᴇʀᴏꜱ ᴀꜱ ᴀʟᴘʜᴀ ᴀɴᴅ ʙᴇᴛᴀ,
ᴛᴏ ꜰɪɴᴅ -->ꜰ(x) ꜱᴜᴄʜ ᴛʜᴀᴛ ɪᴛꜱ ᴢᴇʀᴏꜱ ᴀʀᴇ (ᴀʟᴘʜᴀ +ʙᴇᴛᴀ)ᴀɴᴅ ᴀʟᴘʜᴀ×ʙᴇᴛᴀ,
ɪɴ ᴘ(x) ꜱᴜᴍ ᴏꜰ ᴢᴇʀᴏꜱ=ᴀʟᴘʜᴀ+ʙᴇᴛᴀ=-ʙ/ᴀ,
ᴀʟᴘʜᴀ+ʙᴇᴛᴀ=-(-5)/2=5/2,
ᴘʀᴏᴅᴜᴄᴛ ᴏꜰ ᴢᴇʀᴏꜱ=ᴀʟᴘʜᴀ×ʙᴇᴛᴀ=ᴄ/ᴀ,
ᴀʟᴘʜᴀ×ʙᴇᴛᴀ=-6/2=-3,
ᴢᴇʀᴏꜱ ᴏꜰ ꜰ(x) ᴀʀᴇ 5/2 ᴀɴᴅ -3,
ꜰ(x)=x²-( ꜱᴜᴍ ᴏꜰ ᴢᴇʀᴏꜱ)x+ ᴘʀᴏᴅᴜᴄᴛ ᴏꜰ ᴢᴇʀᴏꜱ,
=x²-(5/2-3)x+(5/2×-3),
=x²-(-1/2)x-(15/2),
=x²+1/2x-15/2,
=2x²+x-15...
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