13. We will now consider certain special equations, the solutions of which depend upon the solutions of quadratic or of linear equations. Certain equations containing two different powers, a2" and ", of an unknown quantity or expression a, one power being the square of the other, may be reduced to the form ax2 + bx2 + c = O which is quadratic with reference to ". Such equations may be solved by the methods employed for the solution of the standard quadratic equation.

2n

E.g. The following equations are all in quadratic form, since in each case the power of the unknown appearing in the first term is the square of the power appearing in the second term :

This equation is equivalent to the set of two quadratic equations

Hence

x2-160 and x2-9 = 0.

x=4 and

x = 13.

(1)

(2)

(3)

All of these values will be found by substitution to satisfy the given equation.

Ex. 2.

Solve x 14x1 +45= 0.

(1)

Observe that x, which appears in the first term, is the square of which appears in the second term.

Factoring with reference to r1, we obtain,

This single equation is equivalent to the set of equations

Squaring both members of each of the equations above,

These values will be found, by substitution, to satisfy the given equation.

To obtain x from æ3, we may raise x3 to the third power and extract the square root of the result, or first extract the square root of x3, and then find the third power of the result.

These values will be found, by substitution, to satisfy the given equation.

Instead of proceeding as above, we may obtain the values as follows:

14. If the solution of a given equation cannot be readily obtained by factoring, we may either resort to the method of completing the square, or we may use the formula.

Observe that in the first term is the square of x in the second term. Referring to the standard equation ax2 + bx + c = 0, we find that x and in the given equation are represented by z2 and x respectively in the standard equation; the coefficient 3 of x in the given equation is represented by the coefficient a of x2 in the standard equation; 8 is repre sented by b, and 2 is represented by c.

Corresponding to the solution of the standard equation,

Squaring both members of each of the equations above,

These exact values will be found by substitution to satisfy the given equation.

By extracting the square root of 10 to any required number of significant figures, and replacing 10 by the approximate value thus found, approxi

mate values may be obtained for x, which are correct to any required number of significant figures.

Thus,

x = 5.6997+,

$ 20, Ex. 2.)

and x = .0780+. (See Chap. XXII.

EXERCISE XXIV. 3

Find one or more solutions of each of the following equations, verifying all integral and fractional solutions:

15. Occasionally the terms of an equation of degree higher than the second may be so grouped as to allow of the reduction of the equation to a form which is quadratic with respect to some definite group of terms containing the unknown.

If we let f(x) represent a group of terms containing the unknown, 2, we may represent an equation which is quadratic with reference to this group of terms by

a[ƒ (x)]2 + b[f(x)] + c = 0.

Ex. 1. Solve (x2 + 3 x)2 - 2(x2+3x) 80.

-

(1)

Observe that the equation is quadratic with respect to the group of termis

(x2 + 3 x).