গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

Given the equations:

1. $x^2 + ax + b = 0$
2. $2x^2 + ax + 16 = 0$

It is given that one of the roots of the equation $2x^2 + ax + 16 = 0$ is $2$.
Substituting $x = 2$ into $2x^2 + ax + 16 = 0$:
$2(2)^2 + a(2) + 16 = 0$
$8 + 2a + 16 = 0$
$2a = -24 \implies a = -12$

Now, the equation becomes:
$2x^2 - 12x + 16 = 0$
$x^2 - 6x + 8 = 0$
$(x - 2)(x - 4) = 0$
Thus, the roots of the second equation are $x = 2$ and $x = 4$.

Since the "other root" of this equation is $2$, the common root between the two equations must be $x = 4$.

Since $x = 4$ is a root of $x^2 + ax + b = 0$:
$(4)^2 + a(4) + b = 0$
$16 + (-12)(4) + b = 0$
$16 - 48 + b = 0$
$b - 32 = 0 \implies b = 32$

Therefore, the required values are:
$a = -12, \quad b = 32$