Locating these points and joining them produces the graph of (1), which consists of two separate branches, CD and EF. Locating two points of equation (2) and joining by a straight line, we have the graph AB of the equation (2). The coördinates of the two points of intersection P and P' are the required roots. By actual measurement we find x = 4.5+, y = 2.5+, or x=2.5, y = — =-4.6. To obtain a greater degree of accuracy, the portion of the diagram near P is represented on a larger scale in the small diagram. Since the small part of CD almost a straight line, it is sufficient to locate 2 or 3 points of this line. which is represented is By actual measurement we find : x = 4.606, y = 2.606. x=2.606, y=4.606.. Evidently the second pair is By increasing the scale further and further, any degree of accuracy may be obtained. 6. Show graphically that the following system cannot have finite roots: (2x-y=2, 7. What are the relative positions of the graphs of two linear inconsistent equations? 8. Show graphically that the following system is satisfied by an infinite number of roots: CHAPTER XVIII QUADRATIC EQUATIONS INVOLVING ONE UNKNOWN QUANTITY 315. A quadratic equation, or equation of the second degree, is an integral rational equation that contains the square of the unknown number, but no higher power; e.g. x2-4x=7, 6 y2=17, ax2 + bx + c = 0. 316. A complete, or affected, quadratic equation is one which contains both the square and the first power of the unknown quantity. 317. A pure, or incomplete, quadratic equation contains only the square of the unknown quantity. ax2 + bx + c = 0 is a complete quadratic equation. ax2 = m is a pure quadratic equation. PURE QUADRATIC EQUATIONS 318. A pure quadratic is solved by reducing it to the form a, and extracting the square root of both members. = Clearing of fractions, ax - x2 - 4 a2 + 4 ax = ax + 4 a2 + x2 + 4 ax. 20. If a2+b2 = c, find a in terms of b and c 21. If d=2, solve for t. 22. If 2 a2 + 2 b2 = 4 m2 + c2, solve for m. 23. Solve the equation of the preceding example for c. 1. Find a positive number which is equal to its reciprocal. . The ratio of two numbers is 5:4, and their product is 980. Find the numbers. 3. Three numbers are to each other as 2:3:4, and the sum of their squares is 261. Find the numbers. 4. Two numbers are as 5:4, and the difference of their squares is 36. Find the numbers. 5. The sides of two square fields are as 7: 24, and they contain together 10,000 square yards. Find the side of each field. 319. A right triangle is a triangle, one of whose angles is a right angle. The side α opposite the right angle is called the hypotenuse (c in the diagram). If the hypotenuse b contains c units of length, and the two other sides respectively a and b units, then c2 = a2 + b2. Since such a triangle may be considered one half of a rectangle, its area contains ab square units. 6. The hypotenuse of a right triangle is 15 inches, and the two other sides are as 3:4. Find the sides. 7. The hypotenuse of a right triangle is to one side as 41: 9, and the third side is 40 centimeters. Find the unknown sides and the area. |