Question

Find the zeros of the quadratic polynomial x^2−5x−36, and verify the relationship between the zeros and the coefficients.

Answer 1

Given

$\sf\pink{⟶}$ Quadratic polynomial

• p(x) = x² - 5x - 36

To find

• Zeroes of the polynomial

To verify

• Relationship between the zeroes and the coefficients.

Solution

$\: \: \: \: \: \: \: \: \: \: \boxed{\tt{\bigstar{By\: middle\: term\: spiliting{\bigstar}}}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {p(x) = x^2 - 9x + 4x - 36}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {p(x) = x(x - 9) + 4(x - 9)}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {p(x) = (x - 9) (x + 4)}$

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★ Zeroes of the polynomial are

$\tt\longmapsto{x = 9}$

$\tt\longmapsto{x = -4}$

⠀

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⠀

$\boxed{\tt{\bigstar{Relationship\: between\: zeroes\: and\: coefficients{\bigstar}}}}$

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1)

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

⠀

$\longrightarrow{(9) + (-4) = \dfrac{-(-5)}{1}}$

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$\longrightarrow{5 = 5}$

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2)

$\longrightarrow\sf{Product\: of\: zeroes = \dfrac{constant}{coefficient\: of\: x^2}}$

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$\longrightarrow{(9) \times (-4) = \dfrac{-36}{1}}$

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$\longrightarrow{-36 = -36}$

⠀

$\underline{\underline{\bf{\orange{Hence\: verified}}}}$

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Answer 2

ɢɪᴠᴇɴ:-

• A Quadratic polynomial:- $\tt P(x) = x^2 - 5x - 36$

ᴛᴏ ꜰɪɴᴅ:-

• Zeroes of the given polynomial

• Relationship between the zeros and the coefficients.

ꜱᴏʟᴜᴛɪᴏɴ:-

✁By splitting the middle term,

$\sf\dashrightarrow \: \: \: \: \: \: \: {p(x) = x^2 - 9x + 4x - 36}$

⠀

$\sf\dashrightarrow\: \: \: \: \: \: \: \: {p(x) = x(x - 9) + 4(x - 9)}$

⠀

$\sf\dashrightarrow\: \: \: \: \: \: \: \: {p(x) = (x - 9) (x + 4)}$

⠀

So,

The zeroes of the given polynomial are 9 and (-4)

$\begin{lgathered}\large \tt Sum \: of \:zeroes = \frac{ - b}{a} \\ \\ \tt \implies {(9) + (-4) = \dfrac{-(-5)}{1}} \\ \\ \tt \implies {5=5} \\ \\ \tt\large product \: of \:zeroes = \frac{ c}{a} \\ \\ \tt \implies {(9) \times (-4) = \dfrac{-36}{1}}\\ \\\tt \implies{ - 36=-36}\end{lgathered}$

Hence,

$\begin{lgathered}\bf\green{Relationship\: between \:the \:zeroes\: and}\:\\\bf\green {coefficients\: is\: \underline{verified}}\end{lgathered}$

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