Example. Find the limits between which a must lie in order that ax2-7x+5 may be capable of all values, a being any real quantity. Put then ax2-7x+5 (a− 5y) x2 - 7x (1 − y) + (5 − ay)=0. In order that the values of x found from this quadratic may be real, the expression that is, 49 (1-y)2 - 4 (a – 5y) (5 – ay) must be positive, (49 – 20a) y2 + 2 (2a2 + 1) y + (49 − 20a) must be positive; hence (2a2+1)2- (49-20a)2 must be negative or zero, and 49 - 20a must be positive. Now (2a2+1)2- (49 – 20a)2 is negative or zero, according as 2 (a2 – 10a +25) × 2 (a2 + 10a − 24) is negative or zero; that is, according as 4 (a -5)2 (a+12) (a − 2) is negative or zero. This expression is negative as long as a lies between 2 and -12, and for such values 49 - 20a is positive; the expression is zero when a=5, – 12, or 2, but 49-20a is negative when a=5. Hence the limiting values are 2 and - 12, and a may have any intermediate value. EXAMPLES. IX. b. 1. Determine the limits between which n must lie in order that the equation 2ax (ax+nc)+(n2-2) c2=0 6. If a, ẞ are roots of the equation x2- px+q=0, find the value of (1) a2 (a2ß-1-B) + B2 (B2a-1-a), (2) (a−p)-4+ (B−p)−4. 7. If the roots of la2+nx+n=0 be in the ratio of p: q, prove that 9. If the roots of the equation ax2+2bx+c=0 be a and ß, and those of the equation Ax2+2Bx+C=0 be a+8 and ẞ+8, prove that 10. Shew that the expression pa2+3x-4 p+3x-4x2 will be capable of all values when x is real, provided that p has any value between 1 and 7. 13. If the roots of ax2+2bx+c=0 be possible and different, then the roots of (a+c) (ax2+2bx+c)=2 (ac - b2) (x2+1) will be impossible, and vice versû. (ax-b) (dx−c) 14. Shew that the expression will be capable of all (bx-a) (cx-d) values when a is real, if a2 – b2 and c2-d2 have the same sign. *122. We shall conclude this chapter with some miscellaneous theorems and examples. It will be convenient here to introduce a phraseology and notation which the student will frequently meet with in his mathematical reading. DEFINITION. Any expression which involves x, and whose value is dependent on that of x, is called a function of x. Functions of x are usually denoted by symbols of the form f(x), F(x), (x). Thus the equation y =ƒ (x) may be considered as equivalent to a statement that any change made in the value of x will produce a consequent change in y, and vice versa. The quantities and y are called variables, and are further distinguished as the independent variable and the dependent variable. An independent variable is a quantity which may have any value we choose to assign to it, and the corresponding dependent variable has its value determined as soon as the value of the independent variable is known. *123. An expression of the form P1x" +P ̧21 +P2x2 + ... + P+P1 where n is a positive integer, and the coefficients P, P1, P2,... ..Pn do not involve x, is called a rational and integral algebraical function of x. In the present chapter we shall confine our attention to functions of this kind. *124. A function is said to be linear when it contains no higher power of the variable than the first; thus ax + b is a linear function of x. A function is said to be quadratic when it contains no higher power of the variable than the second; thus ax2 + bx + c is a quadratic function of x. Functions of the third, fourth,... degrees are those in which the highest power of the variable is respectively the third, fourth,.... Thus in the last article the expression is a function of x of the nth degree. *125. The symbol ƒ(x, y) is used to denote a function of two variables x and y; thus ax + by + c, and ax2+ bxy + cy2 + dx + ey + ƒ are respectively linear and quadratic functions of x, y. The equations f(x) = 0, ƒ (x, y) = 0 are said to be linear, quadratic,... according as the functions f(x), ƒ (x, y) are linear, quadratic,.... *126. We have proved in Art. 120 that the expression ax2+ bx + c admits of being put in the form a (x-a) (x - ẞ), where a and ẞ are the roots of the equation ax2 + bx + c = 0. Thus a quadratic expression ax + bx + c is capable of being resolved into two rational factors of the first degree, whenever the equation ax2 + bx + c = 0 has rational roots; that is, when b2-4ac is a perfect square. *127. To find the condition that a quadratic function of x, y may be resolved into two linear factors. Denote the function by f(x, y) where f(x, y) = ax2+2hxy+by+2gx+2fy+c. thus or Write this in descending powers of x, and equate it to zero; ax2 + 2x (hy + g) + by3 + 2ƒy + c = 0. Solving this quadratic in x we have x= − (hy+g)=√(hy + g)* − a (by* + 2ƒy + c) a ax+hy+g=± √ y3 (h2 — ab) + 2y (hg — aƒ) + (g3 — ac). Now in order that f(x, y) may be the product of two linear factors of the form pa+qy+r, the quantity under the radical must be a perfect square; hence (hg — af )3 =- (h2 — ab) (g3 — ac). Transposing and dividing by a, we obtain abc+2fgh-af-bg" - ch2 = 0; which is the condition required. This proposition is of great importance in Analytical Geometry. *128. To find the condition that the equations ax2 + bx + c = 0, a'x2 + b'x + c = 0 may have a common root. Suppose these equations are both satisfied by xa; then To eliminate a, square the second of these equal ratios and equate it to the product of the other two; thus It is easy to prove that this is the condition that the two quadratic functions ax + bxy + cy3 and a'x2 + b'xy + c'y may have a common linear factor. *EXAMPLES. IX. c. 1. For what values of m will the expression y2+2.xy+2x+my - 3 be capable of resolution into two rational factors? 2. Find the values of m which will make 2x2+mxy+3y2 - 5y-2 equivalent to the product of two linear factors. have a common root, shew that it must be either 5. Find the condition that the expressions lx2+mxy+ny2, l'x2+m'xy+n'y2 may have a common linear factor. 6. If the expression 3x2+2Pxy+2y2+2ax−4y+1 can be resolved into linear factors, prove that P must be one of the roots of the equation P2+4aP+2a2+6=0. 7. Find the condition that the expressions ax2+2hxy+by2, a'x2+2h'xy+b'y2 may be respectively divisible by factors of the form y − mx, my +x. 8. Shew that in the equation x2 - 3xy + 2y2 — 2x-3y - 35=0, for every real value of x there is a real value of y, value of y there is a real value of x. and for every real 9. If x and y are two real quantities connected by the equation 9x2 + 2xy + y2 - 92x-20y+244-0, then will x lie between 3 and 6, and y between 1 and 10. 10. If (ax2+bx+c) y+a'x2+b'x+c=0, find the condition that a may be a rational function of y. |