The reduction to the above form is made: 1st. By transposing all the terms involving x2 and x first member, and all the known terms to the second. to the 2d. If the term involving 2 should be negative, the signs of all the terms of the equation must be changed to render it positive, and then divide both members by the co-efficient of x2. Hence, every complete equation of the second degree can be reduced to an equation involving but three terms, and of the above form. The quantity q is called the absolute term. If we compare the first member of the equation x2 + 2px = 9, with the square of a binomial (x + a)2 = x2 + 2ax + a2 we see that it needs but the square of p to render it a perfect square. If then, p2 be added to the first member, it will become a perfect square; but in order to preserve the equality of the members, p2 must also be added to the second member. Making these additions, we have x2 + 2px + p2 = q + p2 ; this is called completing the square, which is done by adding the square of half the co-efficient of x to both members of the equa tion. Now, if we extract the square root of both members, we have, x + p = ± √ q + p2, and by transposing p, we shall have Either of these values being substituted for x in the equation x2 + 2px = 9 will satisfy it. For, we have from the first value, and x2 = (− p + √ q + p2)2 = p2 - 2p √ q + p2 + q + p2 For the second value, we have x2 = ( − p − √ q + p2)2 = p2 + 2p √ q + p2 +q+p2 and 2px = 2p (− p −√√ q+ p2) = − 2p2 − 2p √ q + p2 ; hence, x2 + 2px = 9; and consequently, the values found above, are roots of the equa tion. In order to refer readily, to either of these values, we shall call the one which arises from using the sign before the radical, the first value of x, or the first root of the equation; and the other, the second value of x, or the second root of the equation. Having reduced a complete equation of the second degree to the form we can write immediately the two values of the unknown quantity by the following RULE. The first value of the unknown quantity is equal to half the coefficient of x taken with a contrary sign, plus the square root of the absolute term increased by the square of half the co-efficient of x. The second value of the unknown quantity is equal to half the co-efficient of x taken with a contrary sign, minus the square root of the absolute term increased by the square of half the co-efficient of x. 1. Let us take as an example the equation 2. As a second example, let us take the equation √2.25 = 2. It often occurs, in the solution of equations, that p2 and q are fractions, as in the above example. These fractions most generally arise from dividing by the co-efficient of x2 in the reduction of the equation to the required form. When this is the case, we readily discover the quantity by which it is necessary to multiply the terms of q, in order to reduce it to the same denominator with p2; after which, the numerators may be added together and placed over the common denominator. After this operation, the denominator will be a perfect square, and may be brought from under the radical sign, and will become a divisor of the square root of the numerator. To apply these principles in reducing the radical part of the values of x, in the last example, we have either of which values being substituted for x in the given equa tion, will satisfy it. 3. What are the values of x in the equation ax2- - ac = COC - bx2. If now, we multiply both terms of q by reduced to a common denominator with p2, 4 (a + b)2° 4 (a + b), it will be and we shall have 4. What are the values of x, in the equations, We observe, that if we multiply both terms of q by 2, and then by 12, that q and p2 will have the same denominator; hence, 5. What are the values of x in the equation In this equation, the term which contains the second power of the unknown quantity, is negative; and since that term already stands in the first member of the equation, it can only be rendered positive by changing the sign of every term of the equation. Doing this, transposing, and dividing by 2. we have 3a 2 36. Hence, 3b. and the root of the radical part is equal to EXAMPLES. 4 4 — a2 — 2x — — 5 |