...surd 5 + \/ 3 Multiplier 5 — \/ 3 These two examples are comprehended under the Rule in Art. 62, the product of the sum and difference of two quantities is equal to the difference of their squares. Ex. 3. Find a multiplier that shall make \/ 5 + V 3 rational. Ex. 4. Find a multiplier that shall make...

...proposed equation will become altogether indeterminate. The numerator, being the product it' the cum and difference of two quantities, is equal to the difference of their square?, !<> wit : I* — (4!-f-4ar)= — 4ac. We see. there!' ire, that 2# is a common factor to the...

...into a binomial surd containing only the square root, may be found by applying the principle, that the product of the sum and difference of two quantities, is equal to the difference of their squares. (Art. 235.) The binomial itself, after the sign which connects the terms is changed from -|- to-, or...

...THEOREMS. 34. The following theorems ought to be committed to memory. (a+b) x (ai) = aO-62. That is, the product of the sum and difference of two quantities is equal to the difference of their squares. Thus we have, (2x + 3a) x (2x-3a) = 4x* — 9a2. And so on to other cases. 35. When a quantity is multiplied...

...c=0, the proposed equation will become altogether indeterminate. The numerator, being the pro Tw : cf the sum and difference of two quantities, is equal to the difference of their aquaies. to wit: b1 — (4*-f-4ac)= — toe. We see, therefore, that Sa is a common factor to tbe.namjr:itor...

...yny3. 20. If a-\-b be multiplied into a — b, the product will be a2 — b2, (Art. 86 ;) that is, 191. The product of the. sum and difference of two quantities, is equal to the. difference of their squares. This is an instance of the facility with which general truths are demonstrated in algebra. If the sum...

...10a262+64. But 0+6 represents the sum of two quantities, and a — 6, their difference ; hence, THEOREM III. The product of the sum and difference of two quantities, is equal to the difference of their squares. EXAMPLES. 1. (5+3)(5—3)=25— 9=16=8X2. 2. (2a+6)(2a— 6)=4a2— b\ 3. (2x+32/)(2x-3y)=4x2-9/. 4....

...86. Prod. 4a2— 962. Multiply 3y — c by 3y — c. Prod. 9y2 — ca. Thus, by inspection, we find the product of the sum and difference of two quantities is equal to the difference of their squares. The propositions included in this article are proved also in geometry. (Art. 14.) We can sometimes...

...c=0, the proposed equation will become altogether indeterminate. The numerator, being the pro-'luri of the sum and difference of two quantities, is equal to the difference of their squares, to wit : tt> — (i*-f-4ae)= — 4ac. We «ee, therefore, that Sa is a common factor to the numerator...

...easily seen by actual multiplication, that (a + b) x (a - 6) = a' - 6", or that the product of t?ie sum and difference of two quantities is equal to the difference of their squares ; a theorem which may be used in the multiplication of such quantities as those in the last example,...