9. Two boys run in opposite directions round a rectangular field, the area of which is an acre; they start from one corner and meet 13 yards from the opposite corner; and the rate of one is § of the rate of the other. Determine the dimensions of the field. 10. A, in running a race with B, to a post and back, met him 10 yards from the post. To make it a dead heat, B must have increased his rate from this point 41 yards per minute; and if, without changing his pace, he had turned back on meeting A, he would have come 4 seconds after him. How far was it to the post? x- - 10 of y, or NOTE. If 2x = the number of yards to the post and back, and y the number of yards A runs a minute, then xy-10y the number of yards B runs a minute. +10 = x+10 11. The fore wheel of a carriage turns in a mile 132 times more than the hind wheel; but if the circumferences were each increased by 2 feet, it would turn only 88 times more. Find the circumference of each. 12. A person has $6500, which he divides into two parts and loans at different rates of interest, so that the two parts produce equal returns. If the first part had been loaned at the second rate of interest, it would have produced $180; and if the second part had been loaned at the first rate of interest, it would have produced $245. Find the rates of interest. CHAPTER XXI. PROPERTIES OF QUADRATICS. 263. Every affected quadratic can be reduced to the form ax2 + bx + c = 0, the solution of which gives the two roots 264. As regards the character of the two roots, there are three cases to be distinguished. I. If b2-4ac is Positive and not Zero. In this case the roots are real and unequal. The roots are real, since the square root of a positive number can be found exactly or approximately. If b2-4ac is a perfect square, the roots are rational; if b2-4ac is not a perfect square, the roots are surds. The roots are unequal, since √b2 - 4ac is not zero. II. If b2-4ac is Zero. In this case the two roots are real and equal, since they both become b 2a III. If b2 - 4 ac is Negative. In this case the roots are imaginary, since they both involve the square root of a negative number. The two imaginary roots of a quadratic cannot be equal, since b2-4ac is not zero. They have, however, the same b real part, 2a' and the same imaginary parts, but with opposite signs; such expressions are called conjugate imaginaries. The expression b2-4ac is called the discriminant of the expression ax2 + bx + c. 265. The above cases may also be distinguished as follows: 266. By calculating the value of b2-4ac, we can determine the character of the roots of a given equation without solving the equation. The roots are real and unequal, and rational. (2) 3x2+7x-1=0. The roots are real and unequal, and are both surds. (3) 4x2-12x+9=0. (5) Find the values of m for which the following equation has its two roots equal: If the roots are to be equal, we must have b2-4ac = 0, or (5 m + 2)2 - 8 m (4 m + 1) = 0. For these values of m the equation becomes 4x2+12x+9= 0, and 4x2 - 4 x + 1 = 0, each of which has its roots equal. Exercise 105. Determine without solving the character of the roots of each of the following equations: Determine the values of m for which the two roots of each of the following equations are equal: 11. (m+1)x2 + (m − 1)x + m +1 = 0. 12. (3m+1) x2 + (2 m + 2) x+m = 0. 13. (m −2) x2 + (m −5) x + 2m — 5 = 0. 14. 2mx2 + x2 — 6 mx − 6x+6m+1=0. 15. mx2+2x2 + 2m = 3mx 9x+10. RELATIONS OF ROOTS AND COEFFICIENTS. 267. Consider the equation x2 — 10x + 24 = 0. Resolve into factors, (x — 6)(x − 4) = 0. The two values of x are 6 and 4; their sum is 10, the coefficient of x with its sign changed; their product is 24, the third term. 268. In general, representing the roots of the quadratic equation ax + bx + c = 0 by r, and r2, we have (§ 263), If we divide the equation ax + bx + c = 0 through by b с 0; this may be a, we have the equation x2+-x+ α b α written x2+px+q=0 where p = 9 α a It appears, then, that if any quadratic equation is made to assume the form x2+px+q= 0, the following relations hold between the coefficients and roots of the equation: (1) The sum of the two roots is equal to the coefficient of x with its sign changed. (2) The product of the two roots is equal to the constant term. Thus, the sum of the two roots of the equation x2-7x+8=0 is 7, and the product of the roots 8. |