CHAPTER IX. THE QUADRATIC. 118. The most general type of a quadratic function of one variable is ax2 + bx + c, and the corresponding equation is ax2 + bx + c =0 . In the equation we may divide through by a; then b (4) which is the quadratic reduced to its simplest form. The roots of this are, by Art. 59, x1 = † ( − p + √ p2 — 4q), and x2 = † ( − p − √p2 — 4 q). On account of the double sign of the root-symbol, √ (Art. 48), both values are included in the one expression x = ± ( − p ± √ p2 — 4q), and this is the solution of (B). and this is the solution of (A) The forms of these solutions should be so mastered that for any quadratic equation, in either of the forms (A) or (B), the solution may be written down at once. Ex. 1. The roots of 3 x2 + 2x 4 = 0 are -- x = f(−2±√4 + 48) = } ( − 1 ± √3). Ex. 2. The roots of 2 x2 3x+2=0 are 119. The double root, or double solution of the quadratic, is frequently of the highest importance as giving an unexpected answer to a problem, and through this answer giving us a clearer idea of the nature of the problem. It is only when a problem admits, in spirit, of a double answer, that it involves the solution of a quadratic. A few examples will make this plain. Ex. 1. A man buys a horse and sells him for $24, thus losing as much per cent as the horse cost in dollars. To find the cost. This solution shows the problem to be to a certain extent indefinite, since there is no way of determining whether the cost of the horse was $40 or $60. Ex. 2. The attraction of a planet varies directly as its mass and inversely as the square of the distance from its centre. The earth's mass is 75 times that of the moon, and their distance apart is 240000 miles. To find a point, in the line joining them, where their attractions are equal. The smaller of these numbers evidently gives EP; the larger, being greater than 240000, gives a second point, Q, beyond the moon, and not contemplated in the problem. Our judgment tells us that there is a second point. Cor. In the foregoing question let the masses of the moon and earth be the same. Then we have That is, one point, P, is half way between the earth and moon, and the other is infinitely distant. EXERCISE IX. a. 1. Solve the quadratics. i. x2+x-1(bc) - bc = 0. ii. x2 iii. abx2 - (a2 + b2)x + ab = ax + (a2 - b2) = 0. = 0. iv. 3x22x + 1 = 0. v. (a2 - b2)x2 - 2 ax + 1 = 0. vi. x2 x = 0. 2. Find the relation between a and b in the equation (a + x)(b − x) + abx − 1 = 0, when i. The sum of the roots is zero. ii. The sum of the reciprocals of the roots is zero. iii. The sum of the reciprocals of the roots is infinite. 3. If the equation ax2 + bx + c = 0 has a and ẞ as its roots, 1 1 find the equation which has and as its roots. α β 4. Show that the roots of ax2 + bx + a = 0 are reciprocals of one another. 5. The area of a right-angled triangle is a2 and the difference between the two sides is d; to find the sides. Explain the double solution, and draw figures to represent it, when a2 = 4 and d = = 2. 6. ABCD is a square. P is a point on AB produced, and Q is on AD, and PCQ is a right angle. Determine BP so that the triangle PCQ shall have a given area, a2. Explain the double solution. 7. In Ex. 6, AQ is equal to BP; determine BP when the triangle PCQ has a given area, a2. Explain the double solution. 8. AB and CD are two straight lines intersecting at right angles in O. AC is of a given length, 7. Find AO when the triangle ACO has a given area, a2. Explain the quadruple solution. 9. Find the area of the triangle of Ex. 8, when AO = 2 Co. 10. Find CO, of Ex. 8, when A021. CO. is the same for each root, the difference in the roots being due to the part √b2 - 4 ac. As this part may be rational, irrational, or imaginary, both roots are alike rational, irrational, or imaginary. (1) When √b2-4ac is real, the roots are real and different. This occurs when a and c have unlike signs, or when they have like signs and b2>4 ac. Ex. 1. The roots of x2 - 2x 2 are 1±√3. Ex. 2. The roots of x2 3x + 1 are 3 ±√5. (2) When √2 - 4ac = 0, the roots are real and equal. This occurs when b2 = 4 ac, in which case the function is a complete square. Ex. 3. The roots of x2 4x4 are 2 ± 0. (3) When √b2-4ac is imaginary, the roots are complex numbers, unless b is zero, when they are imaginaries. This occurs when b2 <4 ac. Ex. 4. The roots of x2-2x+2 are 1 i. (4) When b= 0, the roots are differ in sign only; but they may be or imaginary. ± 1√—4ac, and 2 a rational, irrational, (5) If the roots are real, and a is +, they will have the same sign when b>√b2 - 4 ac; that is, when c is +. The sign of the roots will be the opposite to that of b. This takes place in Ex. 2, the roots being real, and a and c being both +. |