Question

find the sum and product of roots of x2 -6x+5 =0​

Answer 1

Answer:

ANSWER

We know that for a quadratic equation ax

2

+bx+c=0, the sum of the roots is −

a

b

and the product of the roots is

a

c

.

Here, the given quadratic equation x

2

−5x+8=0 is in the form ax

2

+bx+c=0 where a=1,b=−5 and c=8.

The sum of the roots is −

a

b

that is:

−

a

b

=−

1

(−5)

=5

The product of the roots is

a

c

that is:

a

c

=

1

8

=8

Hence, sum of the roots is 5 and the product of the roots is 8.

Answer 2

$\textbf {Sum of the roots = 6 }$

$\textbf{Product of the roots = 5 }$

Given :

$x^{2} -6x+5=0$

To Find :

$\text{Sum of the roots}$

$\text{product of the roots}$

Formula Applied :

$\text{Sum of the roots }=\alpha + \beta = \frac{-b}{a}$

$\text{Product of the roots}=\alpha \beta = \frac{c}{a}$

Solution :

$\text{Quadratic Equation }= ax^{2} + bx+c=0$

$\text{Given equation }=x^{2} -6x+5 = 0$

$\implies \text{ a = coefficient of x^{2} }\implies 1$

$\implies \text{b = coefficient of x} \implies -6$

$\implies \text{c = constant value }\implies 5$

Hence we know that,

$\implies \text{Sum of the roots} = \alpha + \beta$

$\implies \alpha +\beta = \frac{-b}{a} = \frac{-(-6)}{1} \implies 6$

$\implies \text{Product of the roots}= \alpha \beta$

$\implies \alpha \beta = \frac{c}{a} = \frac{5}{1} \implies 5$

Explanation :

Let α and β are the roots of the equation $ax^{2} +bx+c=0$ then,

$\alpha = \frac{-b+\sqrt{b^{2} -4ac} }{2a} , \beta = \frac{-b-\sqrt{b^{2} - 4ac} }{2a}$

$\alpha + \beta = \frac{-b+\sqrt b^{2}-4ac -b - \sqrt{b^{2}-4ac } }{2a}$

$\alpha + \beta = \frac{-b-b}{2a} \implies \frac{-2b}{2a} \implies \frac{-b}{a}$

$\alpha \beta = (\frac{-b+\sqrt{b^{2} -4ac} }{2a} )(\frac{-b-\sqrt{b^{2} -4ac } }{2a} )$

$\alpha \beta \implies \frac{b^{2}+b\sqrt{b^{2} -4ac }\: - b\sqrt{b^{2}-4ac } \: - (\sqrt{b^{2} -4ac} )^{2} }{4a^{2} }$

$\alpha \beta= \frac{b^{2}-(\sqrt{b^{2}-4ac} )^{2} }{4a^{2} } \implies \frac{b^{2}-(b^{2} -4ac)}{4a^{2} }$

$\alpha \beta = \frac{b^{2} -b^{2}+4ac }{4a^{2} } \implies \frac{4 ac}{4a^{2} } \implies \frac{c}{a}$

Since $(x-\alpha )$ and $(x-\beta )$ are the factors of $ax^{2} - bx +c =0,$

We have      $(x-\alpha )(x-\beta )=0$

Hence, $x^{2} -(\alpha+\beta )x + \alpha \beta =0$

That is, $x^{2} - \textbf{(sum of the roots)}x + \textbf{product of the roots}=0$ is the

general form of the quadratic equation when the roots are given.