Question

What is/are the condition(s) for $(a^2-4)x^{d-2}+bx+c=0$ to be a quadratic equation in $x$?

• A. $a$ can be ANY real number other than 2, -2
• B. $a,b,c$ are ALL real numbers where $a$ cannot be 4 or -4
• C. $d=4$
• D. $a,b,c$ are real

Answer 1

Answer: (A), (C), (D)

The correct options are
A

$a$ can be ANY real number other than 2, -2

C

$d=4$

D

$a,b,c$ are real

The standard form of any quadratic equation is $a{x}^2+bx+c=0$, provided $a,b,c$ [coefficients] are real numbers and $a\ne 0$ [$a$ is called the leading coefficient]. So, in the above question, let us look at the conditions one by one.

1) Coefficients should be real - which means, $(a^2-4)$, $b$ and $c$ should be real which follows that $a,b$ and $c$ are real. [If $(a^2-4)$ is real, this follows that $a^2$ is real and hence, $a$ is real]

2) The degree of the polynomial should be 2 - in this case, the degree is given by $d-2$. So, $d-2=2$, which means $d=4$.

3) Leading coefficient is non-zero a real number.
So, $(a^2-4)$ should not be zero.
$a^2-r\ne 0$
$a^2\ne 4$
$a\ne 2,-2$

So, the correct options are A, C and D.