Quadratic equation form?

Question

Suppose we know that the sum of two positive numbers is $2k$ and their product is $m$ then which of the following will be its quadratic equation and why?

1) $x^2$+ $(2k)x$+ $m$= $0$

2) $x^2$- $(2k)x$+ $m$= $0$

Answer 1

Let the two numbers be $x, y$.

Then $x + y = 2k$, and $xy = m$.

Now, $x+y = 2k\implies y = 2k - x$.

Substitute $y$ into the equation $xy = m$ to obtain a quadratic in $x$:

$$x y = m \iff x(2k - x) = m$$

Now, manipulate this equation to obtain the form $ax^2 + bx + c = 0$:

$$x(2k - x) = m \iff 2kx - x^2 = m \iff x^2 - (2k)x + m = 0$$

Answer 2

If $\alpha$ and $\beta$ are the roots of a monic quadratic $x^2+bx+c$, then

$x^2+bx+c=(x-\alpha)(x-\beta)=x^2-(\alpha+\beta)x+\alpha \beta$.

In general $b=-(\alpha+\beta)$ and $c=\alpha \beta$ by equating coefficients.

If $\alpha+\beta=2k$, and $\alpha\beta=m$, subbing in gives

$x^2-(2k)x+m$