465. We shall now discuss some peculiarities which may arise in the solution of a quadratic equation.
Let the equation be
ax^2 + bx+c=0.
If c=0, then ax^2 + bx=0; whence x=0, x = -b a, that is, one of the roots is zero and the other is finite.
If b=0, the roots are equal in magnitude and opposite in sign.
If a=0, the equation reduces to be bx+c=0; and it appears that in this case the quadratic furnishes only one root, namely, -c b. But every quadratic equation has two roots, and in order to discuss the value of the other root we proceed as follows:
Write 1 y for x in the original equation and clear of fractions ; thus, cy^2 + by +a=0. Now put a= 0, and we have cy^2+ by=0; the solution of which is y=0, or -b c, that is, x=\infty, or -c b
Hence, in any quadratic equation one root will become infinite if the coefficient of x becomes zero.
This is the form in which the result will be most frequently met with in other branches of higher Mathematics, but the student should notice that it is merely a convenient abbreviation of the following fuller statement:
In the equation ax^2 + bx+c=0, if a is very small, one root is very large, and as a is indefinitely diminished this root becomes indefinitely great. In this case the finite root approximates to - c b as its limit.
