Question

Find the zeros of the quadratic polynomial 5x^2+12x+7and verify the relationship between the zeros and the coefficients

Answer 1

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Quadratic eqn:- $\bold{ </strong><strong>5 {x}^{2} + 12x + 7</strong><strong>}$

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⭐Relation between their zeros and coefficient.

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Let a = 5 , b = 12 and c = 7

Let $\</strong><strong>a</strong><strong>l</strong><strong>p</strong><strong>h</strong><strong>a</strong><strong>$ and $\</strong><strong>b</strong><strong>e</strong><strong>t</strong><strong>a</strong><strong>$ be the roots of the given quadratic equation.

By using Middle splitting term,

$\implies$ $\bold{</strong><strong>5 {x}^{2} + 12x + 7</strong><strong>}$

$\implies$ $\bold{ </strong><strong>5 {x}^{2} + 5x + 7x + 7</strong><strong>=</strong><strong> </strong><strong>0</strong><strong>}$

$\implies$ $\bold{</strong><strong>5x( x + 1) + 7(x + 1)</strong><strong> </strong><strong>=</strong><strong> </strong><strong>0</strong><strong>}$

$\implies$ $\bold{</strong><strong>(5x + 7)(x + 1) = 0</strong><strong> }$

$\implies$ $\bold{</strong><strong>x = \frac{ - 5}{7} or \: x = - 1</strong><strong> }$

Now, $\</strong><strong>a</strong><strong>l</strong><strong>p</strong><strong>h</strong><strong>a</strong><strong> </strong><strong>=</strong><strong> </strong><strong>\frac{ - 5}{7}</strong><strong>$ and $\</strong><strong>b</strong><strong>e</strong><strong>t</strong><strong>a</strong><strong> </strong><strong>=</strong><strong> </strong><strong>-</strong><strong>1</strong><strong>$

The relationship between sum of zeroes and coefficient of Quadratic polynomial is given by :-

★ $\boxed{\sf{ </strong><strong>\alpha + \beta = \frac{ - b}{a} </strong><strong>}}$

$\implies$ $\bold{ </strong><strong>\frac{ - 5}{7} + 1 = \frac{ - 12}{5} </strong><strong>}$

$\implies$ $\bold{</strong><strong> \frac{ - 7 - 5}{ 5} </strong><strong>=</strong><strong> </strong><strong>\frac{ - 12}{5} </strong><strong> }[/tex</strong><strong>]</strong></p><p></p><p>[tex]\implies$ $\bold{ \frac{-12}{5} = \frac{-12}{5} }$

L. H. S = R. H. S verified.

Now,

Relation between products of zeroes and their coefficient is given by :-

★ $\boxed{\sf{</strong><strong> \alpha \beta = \frac{c}{a} </strong><strong> }}$

$\implies$ $\bold{</strong><strong> \frac{ - 7}{5} \times - 1 = \frac{7}{5} </strong><strong>}$

$\implies$ $\bold{</strong><strong> \frac{7}{5} = \frac{7}{5} </strong><strong>}$

L. H. S = R. H. S hence ,verified..