Pure Quadratic Equations.

161. Examples.

1. Solve the equation $5x^2-48=2x^2$.

lint We have 5x^2 - 48 = 2x^2. lint Collect the terms, 3x^2 = 48. [1] lint Divide by $3$, x^2 = 16. lint Extract the square root, x = ±4.

The sign $\pm$ before the $4$, read plus or minus, shows that the root is either $+$ or $-$. For $(+4)\times (+4)=16$, and $(-4)\times (-4)=16$

The square root of any number is positive or negative. Hitherto we have given only the positive value. In this chapter we shall give both values. This sign $\surd$, called the radical sign, is used to indicate that a root is to be extracted. Thus $\sqrt{4}$ means the square root of $4$ is required. $\sqrt[3]{4}$ means the third root of $4$ is required; the small figure placed over the radical sign is called the index of the root, and shows the root required.

2. Solve the equation $3x^2-15=0$.

lint We have 3x^2 = 15, lintor x^2 = 5. lint Extract the square root, x = ±5.

The roots cannot be found exactly, since the square root of $5$ cannot be found exactly; it can, however, be determined approximately to any required degree of accuracy; for example, the roots lie between $2.23606$ and $2.23607$; and between $-2.23606$ and $-2.23607$.

3. Solve the equation $3x^2+15=0$.

lint We have 3x^2 = -15, lintor x^2 = -5. lint Extract the square root, x = ±-5.

There is no square root of a negative number, since the square of any number, positive or negative, is positive; $(-5)\times (-5)=+25$.

The square root of $-5$ differs from the square root of $+5$ in that the latter can be found as accurately as we please, while the former cannot be found at all.

A root which can be found exactly is called an exact or rational root. Such roots are either whole numbers or fractions.

A root which is indicated but can be found only approximately is called a surd. Such roots involve the roots of imperfect powers.

Rational and surd roots are together called real roots.

A root which is indicated but cannot be found, either exactly or approximately, is called an imaginary root. Such roots involve the even roots of negative numbers.