Question

find the zeros of quadratic polynomial and verify the relationship between zeroes and coefficients p(x)=t^2 +7t+12​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

⠀⠀⠀$\huge\underline{ \mathrm{ \red{QueS{\pink{tiOn}}}}}$

find the zeros of quadratic polynomial and verify the relationship between zeroes and coefficients p(x)=t^2 +7t+12

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$\large\underline{ \underline{ \red{ \bold {Answer}}}}$

verified

⠀⠀⠀$\huge{ \underline{ \purple{ \bold{ \underline{ \mathrm{ExPlanA{\green{TiOn }}}}}}}}$

$\bf{ \color{purple}{Given}\begin{cases}\textsf{p(x)= t square +7t+12}\end{cases}}$

$\large\underline{ \underline{ \red{ \bold {To \:Find}}}}$

we need to find the zeroes of the given polynomial and relationship between the zeroes and coefficients.

⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \orange{ \bold{sOluTiOn}}}}$

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$\underbrace{ \color{red}{ \bf{by \: splitting \: the\: middle\:term}}}$

➩⠀⠀⠀⠀⠀$\bf\:{t}^{2}+7t+12$

➩⠀⠀⠀⠀⠀$\bf\:{t}^{2}+4t+3t+12$

➩⠀⠀⠀⠀⠀$\bf\:t(t+3)+3(t+4)$

➩⠀⠀⠀⠀⠀$\bf\:(t+3)(t+4)$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀${\boxed{\color{red}{\bf{t=-3\:and\:t=-4}}}}$

$\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: let \: \\ \: \alpha = - 3 \\ \bf\beta = - 4$

Relationship between zeroes and coefficients:-

compare $\bf\:{t}^{2}+7t+12=0$with $\bf\:a{x}^{2}+bx+c=0$

• a = 1

• b = 7

• c = 12

${\underline{\boxed{ \color{green} \bf{sum \: of \: zeroes = \frac{coefficient \: of \: x}{coefficient \: of \: {x}^{2} } }}}}$

$\bf \alpha + \beta = \frac{ - b}{a}$

➩⠀⠀⠀⠀⠀$\bf -3 + (-4) = \frac{ -7}{1}$

➩⠀⠀⠀⠀⠀$\bf -3-4 = -7$

➩⠀⠀⠀⠀⠀$\bf -7= -7$

⠀⠀⠀⠀⠀⠀⠀⠀${\underline{\boxed{\purple {\bf{LHS=RHS } }}}}$

Hence verified.

$\small {\underline{\boxed{ \color{green} \bf{Product \: of \: zeroes = \frac{constant \:term}{coefficient \: of \: {x}^{2} } }}}}$

$\bf \alpha \times \beta = \frac{ c}{a}$

➩⠀⠀⠀⠀⠀$\bf -3\times-4 = \frac{12}{1}$

➩⠀⠀⠀⠀⠀$\bf 12 = 12$

⠀⠀⠀⠀⠀⠀⠀⠀${\underline{\boxed{\purple {\bf{LHS=RHS } }}}}$

Hence verified.

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