It has been shown (Art. 138), that every equation of the second degree can be reduced to the form Р 'x2+px=q.... (1), and q being numerical or algebraic quantities, whole numbers or fractions, and their signs plus or minus. 202 4 If, in order to render the first member a perfect square, we add to both members, the equation becomes Whatever may be the value of the number expressed by q+ its root can be denoted by m, and the equation becomes p2 But as the first member of this equation is the difference between two squares, it can be put under the form in which the first member is the product of two factors, and the second is 0. Now we can render the product equal to 0, and consequently satisfy the equation (2), in two different ways: viz. Now, either of these values, being substituted for x in its corresponding factor of equation (2) will satisfy that equation; and as equation (1) will always be satisfied when the derived equation (2) is satisfied, it follows, that either value will satisfy equation (1). Hence we conclude, 1st. That every equation of the second degree has two roots, and only two. 2d. That every equation of the second degree may be decomposed into two binomial factors of the first degree with respect to x, having x for a common term, and the two roots, taken with their signs changed, for the second terms. For example, the equation x2+3x-28=0 being resolved gives x=4 and x=—' -7; either of which values will satisfy the equation. We also have (x-4) (x+7)=x2+3x-28. 143. If we designate the two roots by x' and x", we have Hence, 1st. The algebraic sum of the two rooto is equal to the coefficient of the second term of the equation, taken with a contrary sign. 2d. The product of the two roots is equal to the second member of the equation, taken also with a contrary sign. REMARK. The preceding properties suppose that the equation has been reduced to the form x2+px=q; that is, Ist. That every term of the equation has been divided by the co-efficient of x2. 2d. That all the terms involving x have been transposed and arranged in the first member, and a2 made positive. 144. There are four forms, under which the equation of the second degree may be written. In which we suppose p and q to be positive. These equations being resolved, give, In order that the value of x, in these equations, may be found, cither exactly or approximatively, it is necessary that the quantity under the radical sign be positive (Art. 126). p2 Now, being necessarily positive, whatever may be the sign. 4 of p, it follows, that in the first and second forms all the values of x will be real. They will be determined exactly, when the quan tity + is a perfect square, and approximatively when it is not so. In the first form, the first value of x, that is, the one arising from taking the plus value of the radical, is always positive; for the Р radical 4+, being numerically greater than the ex 2 pression -2 ·±√ q+2 p2 is necessarily of the same sign as 4 that of the radical. For the same reason, the second value is essentially negative, since it must have the same sign as that with which the radical is affected: but each root, taken with its proper sign, will satisfy the equation. The positive value will, in general, alone satisfy the problem understood in its arithmetical sense; the negative value, answering to a similar problem, differing from the first only in this; that a certain quantity which is regarded as additive in the one, is subtractive in the other, and the reverse. In the second form, the first value of x is also positive, and the second negative, the positive value being the greater. In the third and fourth forms, the values of x will be imaginary when real values of x will both be negative in the third form, and both positive in the fourth. 145. The same general consequences which have just been remarked, would follow from the two properties of an equation of the second degree demonstrated in (Art. 143). The properties are: The algebraic sum of the roots is equal to the co-efficient of the second term, taken with a contrary sign, and their product is equal to the second member, taken also with a contrary sign. For, in the first two forms, q being positive in the second member, it follows that the product of the two roots is negative: hence, they have contrary signs. But in the third and fourth forms q being negative in the second member, it follows that the product of the two roots will be positive: hence, they will have like signs, viz. both negative in the third form, where p is positive, and both positive in the fourth form where Ρ is negative. Moreover, since the sum of the roots is affected with a sign contrary to that of the co-efficient p; it follows, that, the negative root will be the greatest in the first form, and the least in the second. 146. We will now show that, when in the third and fourth forms, we have the conditions of the question will be incompa 4 tible with each other, and therefore, the values of x ought to be imaginary. Before showing this it will be necessary to establish a proposition on which it depends: viz. If a given number be decomposed into two parts and those parts multiplied together, the product will be the greatest possible when the parts are equal. Let p be the number to be decomposed, and d the difference of the parts. Then Now it is plain that P will increase as d diminishes, and that it will be the greatest possible when d=0: that is, p p3 is the greatest product. 147. Now, since in the equation x2-px=-q p is the sum of the roots, and q their product, it follows that q can |