20. A person after paying a poor-rate, and also an income-tax of 10d. in the pound, has £440 left. The poor-rate is £44 more than the income-tax. Find the original income and the number of pence in the pound of the poor rate. 21. A person was offered some books for £50; but he obtained a discount of 3d. in the shilling on the new books and 1d. in the shilling on the old. The new books on this arrangement cost £4 3s. 4d. more than the old. How much money was spent on each class of books? 22. There is a rectangular floor such that if it had been two feet longer and one foot narrower its area would have been the same; also if it had been four feet shorter and three feet broader its area would have been the same. Find the dimensions of the floor. 23. A person walking along the road in a fog meets one waggon and overtakes another, which is travelling at the same rate as the former, and he observes that between the time of his first seeing and passing the waggons, he walks 20 and 60 yards respectively. Find how far he can see in the fog, and, supposing him to be walking at the rate of 4 miles an hour, find the rate at which each waggon is moving. 24. A railway train after travelling 50 miles is delayed 10 minutes by an accident, after which it proceeds at two-thirds of its former rate and arrives at its destination 1 hour 20 minutes late. Had the detentiou occurred 10 miles further on the train would have arrived 10 minutes sooner than it did. Find the original rate of the train and the distance travelled. 25. A and B run a mile race round a square enclosure PQRS starting from P; and it is observed that when A is at S, B is at R, in the first circuit; also that A completes the race 3 minutes before B. Find the speed of each in miles per hour. PURE QUADRATICS. A quadratic equation is one which contains the square of the unknown quantity, but no higher power. of that quantity. If the first power of the unknown quantity be likewise present the equation is called an adfected quadratic; but if the second power only is found it is called a pure quadratic. = Thus 7 is a pure quadratic; 3x2+2x=5 is an adfected quadratic. A pure quadratic is solved like a simple equation with the exception of the last step, which is completed by extracting the square root of each side. Thus let 3 = to find x. 2 By clearing off fractions, 2(x2+8)=3(x2-3) Thus far the work proceeds as in simple equations. We now extract the square root of each side and x=+5. The reason why + or -5 is taken for the answer is this, viz. (+5) ×(+5)=25 and (−5) × (−5)=25. So that the square root of 25 may be either + or -5, and unless it is clear which of these is to be taken in any given case we write +5 as the answer. ac 4 XC 9 Solve + = + clearing off fractions, ADFECTED QUADRATICS. 1. Before proceeding to the solution of adfected quadratics it will be desirable to recall the rule for squaring a binomial, such as a+b. We know the result to be (a+b)2=a2+2ab+b2, which is obtained by squaring each term and adding twice their product. So that if we have a quadratic expression, such as a2+2ab+b2, consisting of the sum of three terms, two of which are the squares of two quantities, while the third is twice the product of those two quantities, then we know such an expression must be the square of these two quantities added together. Thus p2+2px+x2 is the square of p+x, for p2 is the square of the first of these quantities, viz. p. x2 is second and 2px is twice the product of p and x. :. √p2+2px+a2=p+x. x. When the term containing twice the product is minus, we have then the square of the difference of the quantities. Thus p2-2px+x2=p—x. Similarly (x2-10x+25)=x-5. As a further exercise the pupil may write down from inspection the square roots of the first 12 expressions in Ex. xxxv. 2. This knowledge of quadratic expressions which are perfect square quantities, will enable us to supply any one of the three terms in such an expression when the other two are given. Thus suppose we have x2+25 and are required to supply the term which will make this expression a perfect square. We see that the two terms given are the squares of x and 5 respectively. But besides the squares of the two terms we need also twice their product to make a complete square. Hence adding twice the product of 5 and x, viz. 10x, we get x2+10x+25, which is the square of x+5. This process is termed completing the square; and it is plain that in this case we might also have completed it by subtracting 10x, thus obtaining a2-10x+25, which is the square of x-5. If, however, the missing term be the first or third there is only one way in which we can complete the square, for these terms are always positive and the quantity obtained must always be added. Thus complete the square of the following expressions : 4 1. a2+12a; 2. x2+ x; 3. x2—ax. In 1. The first term is the square of a. The second term is twice the product of 6 and a. Hence, The third term required must be the square of 6. Adding the square of 6 we have a2+12a+62=a2+12a+36, which is the square of a +6. Looking at this more closely we see that since the second term 12a is twice the product of the two quantities squared, their product must be 12a divided by 2 or 6a. And since from the a2 we know the first quantity is a, the second must be 6. Hence the square of 6-36 is the term to be added to complete the square. Therefore we have the following rule for completing the square of an expression, such as a2+ma. Take half the coefficient of a, add the square of the quotient and the result will be a perfect square. 4 Thus in (2) x2+5. 2 .. the completed square is a2+4x+(?)* 2 which is the square of x+5· In 3. x2-ax.-the coefficient of x is a, which As a further exercise the pupil may complete the squares of the expressions in Ex. xxxv., from 13 to 24 inclusive. When these exercises are understood we are then in a position to solve an adfected quadratic equation. 3. Suppose we are to solve the equation x2-4x=5. We can now make 2-4 a perfect square by adding square of half the coefficient of x. the .. Completing the square x2-4x+22=5+4 and x2-4x+22=9. The 4 is added to the right hand side because 22 was |