Question

find the relationship between zeros and coefficients 2x^2+7x+6=0
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Answer 1

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Answer 2

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The quadratic polynomial 2x² + 7x + 6 = 0.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The relationship between zeroes and coefficient.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We have p(x) = 2x² + 7x + 6

Zero of the polynomial p(x) = 0

So;

$\longrightarrow\rm{2x^{2} +7x+6=0}\\\\\longrightarrow\rm{2x^{2} +4x+3x+6=0}\\\\\longrightarrow\rm{2x(x+2)+3(x+2)=0}\\\\\longrightarrow\rm{(x+2)(2x+3)=0}\\\\\longrightarrow\rm{x+2=0\:\:\:Or\:\:\:2x+3=0}\\\\\longrightarrow\rm{x=-2\:\:\:Or\:\:\:2x=-3}\\\\\longrightarrow\rm{\orange{x=-2\:\:\:Or\:\:\:x=-3/2}}$

∴ The α = -2 and β = -3/2 are the zeroes of the polynomial.  

As the given quadratic polynomial as we compared with ax² + bx + c;

• a = 2
• b = 7
• c = 6

Now;

$\underline{\green{\mathcal{SUM\:OF\:THE\;ZEROES\::}}}$

$\mapsto\rm{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } }\\\\\\\mapsto\rm{-2+\bigg(-\dfrac{3}{2}\bigg)=\dfrac{-7}{2} }\\\\\\\mapsto\rm{-2-\dfrac{3}{2} =\dfrac{-7}{2} }\\\\\\\mapsto\rm{\dfrac{-4-3}{2}=\dfrac{-7}{2} }\\\\\\\mapsto\rm{\orange{\dfrac{-7}{2} =\dfrac{-7}{2} }}$

$\underline{\green{\mathcal{PRODUCT\:OF\:THE\;ZEROES\::}}}$

$\mapsto\rm{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:x^{2} } }\\\\\\\mapsto\rm{-2\times \bigg(-\dfrac{3}{2}\bigg)=\dfrac{6}{2} }\\\\\\\mapsto\rm{\cancel{\dfrac{6}{2}}=\cancel{\dfrac{6}{2} }}\\\\\\\mapsto\rm{\orange{3=3}}$

Thus;

Relationship between zeroes and coefficient is verified .