Question

Find a Quadratic polynomial the sum and product of whose zeroes are -5 and 6.​ class 10 please make step by step​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the required polynomial be f(x) having

$\sf \: zeroes \: \alpha \: and \: \beta$

According to statement,

It is given that,

Sum of the zeroes of f(x) = - 5

$\bf\implies \: \alpha + \beta = - \: 5$

and

Product of zeroes of f(x) = 6

$\bf\implies \: \alpha \beta = 6$

We know,

The quadratic polynomial is given by

$\rm :\longmapsto\:f(x) = k\bigg( {x}^{2} - ( \alpha + \beta )x + \alpha \beta \bigg) , \: where \: k \ne \: 0$

So, on substituting the values, we get

$\rm :\longmapsto\:f(x) = k\bigg( {x}^{2} - ( - 5)x +6 \bigg) , \: where \: k \ne \: 0$

$\rm :\longmapsto\:f(x) = k\bigg( {x}^{2} + 5x +6 \bigg) , \: where \: k \ne \: 0$

Additional Information :-

$\rm :\longmapsto\: \alpha , \: \beta \: are \: the \: zeroes \: of \: {ax}^{2} + bx + c \: then$

$\boxed{\purple{\tt Product\ of\ the\ zeroes=\frac{c}{a}}}$

or

$\blue{ \boxed{\bf \: \alpha \beta = \frac{c}{a}}}$

and

$\boxed{\purple{\tt Sum\ of\ the\ zeroes=\frac{-b}{a}}}$

or

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

$\rm :\longmapsto\: \alpha ,\beta, \gamma \: are \: the \: zeroes \: of \: {ax}^{3} + b {x}^{2} + cx + d \: then$

$\blue{ \boxed{\bf \: \alpha + \beta + \gamma = - \: \frac{b}{a}}}$

$\blue{ \boxed{\bf \: \alpha \beta + \beta \gamma + \gamma \alpha = \: \frac{c}{a}}}$

$\blue{ \boxed{\bf \: \alpha \beta \gamma = - \: \frac{d}{a}}}$