CHAPTER XX.

SIMULTANEOUS QUADRATICS.

311. Quadratic equations involving two unknown numbers require different methods for their solution, according to the form of the equations.

CASE 1.

312. When one of the equations is a simple equation.

The original equations are both satisfied by either pair of values. But the values x = 1, y = 7, will not satisfy the equations; nor will the values x = - 5, y = — 1.

The student must be careful to join to each value of x the corresponding value of y.

CASE 2.

313. When the left side of each of the two equations is homogeneous and of the second degree.

Let yvx and substitute vx for y in both equations.

(1)

(2)

CASE 3.

314. When the two equations are symmetrical with respect to x and y; that is, when x and y are similarly involved.

Thus, the expressions

2x3 +3x2y2+2y3, 2xy-3x-3y+1, xa— 3x2y — 3xy2+ya

are symmetrical expressions. In this case the general rule is to combine the equations in such a manner as to remove the highest powers of x and y.

To remove x4 and y1, raise (2) to the fourth power,

x+4x3y+6x2y2 + 4 xy3 + y1

Add (1),

24

= 2401 + y2 = 337

2x4 + 4x3y + 6 x2y2 + 4 xy3 + 2 y1 = 2738

Divide by 2, x2 + 2 x3y + 3x2y2 + 2 xy3 + y = 1369.

Extract the square root,

(1) (2)

Subtract (3) from (2)2, xy 12 or 86.

We now have to solve the two pairs of equations,

315. The preceding cases are general methods for the solution of equations that belong to the kinds referred to; often, however, in the solution of these and other kinds of simultaneous equations involving quadratics, a little ingenuity will suggest some step by which the roots may be found more easily than by the general method.