The first member is now a complete square. Taking the square root of both members, we have whence by transposition, x+a=±√ a2+b; x = −a±√a3+b. Thus the equation has two roots, which are unequal in all cases except when a+b=0; in this case we shall have and the equation is said to have two equal roots. Thus take the equation, Hence, for the solution of an affected quadratic equation we have the following RULE. I. Reduce the given equation to the form of x'+2ax=b. II. Add to both sides of the equation the square of one half the coefficient of x, and the first member will be a complete square. III. Extract the square root of both members, and reduce the resulting equation. 290. When the equation has been reduced to the form of x2+2ax=b, its roots may be obtained directly by the following obvious rule: Write one half the coefficient of x with its contrary sign, plus or minus the square root of the second member increased by the square of one half this coefficient. 1. Given x2-6x = 55, to find the values of x. OPERATION. x=3±√55+9; or, x= =3±8 =11 or -5, Ans. EXAMPLES FOR PRACTICE. Find the values of the unknown quantity in each of the following 291. It frequently happens in reducing a quadratic to the form of x2+2axb, that 2a, the coefficient of x, becomes fractional, thus rendering the solution a little complicated. In such cases it will be sufficient to reduce the first member to the simplest entire The equation will then be in the form terms. in which a and b are integral in form, and prime to each other, and c is entire or fractional. To render the first term of equation (1) a perfect square, multiply both members by a; thus, where the first member is a complete square. Now if b is even, 4 b will be entire; but if b is odd, will be fractional, a result which 4 we wish to avoid. To modify the rule to suit the latter case, suppose equation (3) to be multiplied by 4; thus, The first member is now a complete square, and its terms are entire. Moreover, we observe that (4) may be obtained directly from (1) by multiplying (1) by 4a, and adding b2 to both sides of the result. Hence, for the second method of completing the square in the first member, we have the following RULE. I. Reduce the equation to the form of ax2+bx = c, where a and b are prime to each other. II. If b is even, multiply the equation by the coefficient of x, and add the square of one half the coefficient of x to both members. III. If b is odd, multiply the equation by 4 times the coefficient of x2, and add the square of the coefficient of x to both members. The above rule may be considered as more general than the first; for if applied to equations in the form of x+2ax=b, the operation will be the same as by the first rule, with the simple modification of avoiding fractions in the first member, when 2a is fractional. 1. Given 5x2-6x= 8, to find the values of x. |