Question

Find the zeroes of the quadratic polynomial 6x2

-5x-14 and verify both relations.​

Answer 1

Answer:

2 and - 7/6

Step-by-step explanation:

Given polynomial :

• 6x² - 5x - 14

We can find the zeroes of the quadratic polynomial by splitting the middle term. To split the middle term, we need two numbers whose sum is (-5) and product is (14 * 6 = 84).

Such two numbers can be (-12) and (7).

Here we go,

$\sf \longrightarrow 6x^2 - 5x - 14 = 0 \\\\\longrightarrow \sf 6x^2 -12x + 7x - 14=p \\\\\longrightarrow \sf 6x(x-2)+7(x-2)=0\\\\\longrightarrow \sf (x-2)(6x +7) = 0\\\\\longrightarrow \sf x = 2 \: or \: x = \frac{-7}{6}$

Therefore, the zeroes of the quadratic polynomial is 2 and -7/6.

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Verification :

We know that,

Sum of zeroes = $\sf \dfrac{-(coefficient \: of \: x)}{coefficient \: of \: x^2}$

$\sf \longrightarrow 2 + \frac{-7}{6} = \frac{-(-5)}{6} \\\\\longrightarrow \sf \frac{12-7}{6}=\frac{5}{6}\\\\\longrightarrow \sf \frac{5}{6}=\frac{5}{6} \: \: \: \: \: \: \bold{[\therefore \: LHS = RHS]}$

Product of zeroes = $\sf \dfrac{Constant \: term}{coefficient \: of \: x^2}$

$\longrightarrow \sf 2 \times (\frac{-7}{6}) = \frac{-14}{6}\\\\\longrightarrow \sf \frac{-14}{6}= \frac{-14}{6}\: \: \: \: \: \: \bold{[\therefore \: LHS = RHS]}$

Hence, it is verified.

Answer 2

${\sf{\underline{\underline{\pink{Solution:-}}}}}$

${ \ 2 \ and \ -}$ $\frac{7}{6}$

$\sf\pink{Step-by-step \ explanation:}$

$\sf\red{Given \ polynomial :}$

• $\sf\green{6x² - 5x - 14}$

ᴡᴇ ᴄᴀɴ ꜰɪɴᴅ ᴛʜᴇ ᴢᴇʀᴏᴇꜱ ᴏꜰ ᴛʜᴇ Qᴜᴀᴅʀᴀᴛɪᴄ ᴘᴏʟʏɴᴏᴍɪᴀʟ ʙʏ ꜱᴘʟɪᴛᴛɪɴɢ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ. ᴛᴏ ꜱᴘʟɪᴛ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ, ᴡᴇ ɴᴇᴇᴅ ᴛᴡᴏ ɴᴜᴍʙᴇʀꜱ ᴡʜᴏꜱᴇ ꜱᴜᴍ ɪꜱ (-5) ᴀɴᴅ ᴘʀᴏᴅᴜᴄᴛ ɪꜱ (14 × 6 = 84).

ꜱᴜᴄʜ ᴛᴡᴏ ɴᴜᴍʙᴇʀꜱ ᴄᴀɴ ʙᴇ (-12) ᴀɴᴅ (7).

$\sf\orange{Here \ we \ go,}$

$\begin{gathered}\sf \longrightarrow 6x^2 - 5x - 14 = 0 \\\\\longrightarrow \sf 6x^2 -12x + 7x - 14=p \\\\\longrightarrow \sf 6x(x-2)+7(x-2)=0\\\\\longrightarrow \sf (x-2)(6x +7) = 0\\\\\longrightarrow \sf x = 2 \: or \: x = \frac{-7}{6}\end{gathered}$

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴛʜᴇ ᴢᴇʀᴏᴇꜱ ᴏꜰ ᴛʜᴇ Qᴜᴀᴅʀᴀᴛɪᴄ ᴘᴏʟʏɴᴏᴍɪᴀʟ ɪꜱ 2 ᴀɴᴅ -7/6.

_______________________

$\sf÷purple{Verification :}$

$\sf\blue{We \ know \ that,}$

$Sum \ of \ zeroes = \sf \dfrac{-(coefficient \: of \: x)}{coefficient \: of \: x^2}$

$\begin{gathered}\sf \longrightarrow 2 + \frac{-7}{6} = \frac{-(-5)}{6} \\\\\longrightarrow \sf \frac{12-7}{6}=\frac{5}{6}\\\\\longrightarrow \sf \frac{5}{6}=\frac{5}{6} \: \: \: \: \: \: \bold{[\therefore \: LHS = RHS]}\end{gathered}$

$Product \ of \ zeroes = \sf \dfrac{Constant \: term}{coefficient \: of \: x^2}$

$\begin{gathered}\longrightarrow \sf 2 \times (\frac{-7}{6}) = \frac{-14}{6}\\\\\longrightarrow \sf \frac{-14}{6}= \frac{-14}{6}\: \: \: \: \: \: \bold{[\therefore \: LHS = RHS]}\end{gathered}$

$\sf\red{Hence \ , it \ is \ verified.}$