Question

find the zeros of the qudratic polynomial x^2+7x +10 and verify the relationship between the zeros and it's coefficient
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Answer 1

Step-by-step explanation:

x^2+2x+5x+10

x(x+2) +5(x+2)

(x+5) (x+2)

x+5=0. x+2=0

x= -5. x= -2

Answer 2

This is verified that zeroes and coefficients of quadratic equation are related to each other

Step-by-step explanation:

We are given with a quadratic equation,

$x^{2} +7x+10=0$

and we have to verify the relationship between zeroes and coefficients of equation.

• Formula used,

According to standard quadratic equation,

$ax^{2} +bx+c=0$

Sum of zeroes  $=\frac{-b}{a}$

Product of zeroes $= \frac{c}{a}$

a is the coefficient of $x^{2}$ , b is for coefficient of $x$ and c is the constant.

• Calculation for zeroes

we have,

$x^{2} +7x+10=0$

we can find zeroes by using factorization method,

we can write $7x$ as $5x+2x$  and $5x*2x=10x^{2}$ .

$x^{2} +5x+2x+10=0$

$x(x+5)+2(x+5)=0$

$(x+2)(x+5)=0$

$x=-2,-5$

Zeroes of quadratic equation are -2, -5.

• Verification of relationship between coefficients and zeroes

We have coefficients of $x^{2} ,x$ and constant ,

$a=1$  ,  $b=7$  and  $c=10$.

1). Sum and product of zeroes by using coefficients,

Sum of zeroes  $=\frac{-b}{a}$                              Product of zeroes $= \frac{c}{a}$

                         $=\frac{-7}{1}$                                                             $=\frac{10}{1}$

                        $=-7$                                                              $=10$

2). Sum and product of zeroes by using calculated zeroes,

let zeroes of equation α and β,

$\alpha=-2$    and   $\beta =-5$.

so the sum of zeroes $=\alpha +\beta$                           product of zeroes $=\alpha *\beta$

                                   $=-2+(-5)$                                                   $=(-2)*(-5)$

                                  $=-2-5$                                                          $=10$

                                 $=-7$

Thus we can see sum and product of zeroes by coefficient and by calculated zeroes are same.

So this is verified they are related to each other.