Question

write the quadratic equation whose sum and product og zeros are m+n and m*n​

Answer 1

Given that sum of zeroes is m+n and the product is mn.

So, we can assume that the zeroes of the equation are m and n, can't we?

So let's write the equation whose roots are m and n.

As m and n are roots,  x - m  and  x - n  are the factors.

Therefore,

$(x-m)(x-n) = x^2-mx-nx+mn=\bold{x^2-(m+n)x+mn}$

So an equation is obtained. That's all!

Okay, let me do it in another method.

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TO REMEMBER...

In a quadratic equation ax² + bx + c = 0,

if the roots are α and β,

then,

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

and

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

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By the above concept, let's find an equation whose sum and product of the zeroes are m+n and mn respectively.

Let it be ax² + bx + c = 0.

Here,

α + β = m + n

and

αβ = mn

Assume that the coefficient of x² in the equation ax² + bx + c = 0 is 1.

I.e., a = 1

The roots are let as α and β.

So,

Sum of the roots,

$\alpha+\beta=-\frac{b}{a}=-\frac{b}{1} = -b=m+n \\ \\ \therefore\ b=-(m+n)$

Product of the roots,

$\alpha\beta=\frac{c}{a}=\frac{c}{1}=c=mn \\ \\ \therefore\ c=mn$

∴ $ax^2+bx+c=0$

can be rewritten as

$\bold{x^2-(m+n)x+mn}$  

The roots are m and n.

We can get another equations by giving any values for a.

If a = 2, we get the equation below,

$\bold{2x^2-2(m+n)x+2mn}$

The roots of this equation is also m and n.

Hope this article may be helpful.

Thank you. Have a nice day. :-)

#adithyasajeevan