square of a binomial, we can then reduce the equation x2 + bx=c. Since bi has in it as a factor the square root of x2, x2 can be the first term of the square of a binomial, and bx the second term of the same square; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the quotient arising from dividing the second term of the square by twice the square root of the first term of the ber, we have (2), an equation whose first member is a perfect square. Extracting the square root of each member of (2), and general expression for the value of x in any equation in the form of x2 + bx = C. Hence, as every affected quadratic equation can be reduced to the form x2+ bx =c, in which b and c represent any quantities whatever, positive or negative, integral or fractional, every affected quadratic equation can be reduced by the following RULE. Reduce the equation to the form x2 + bx=c, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. must have two roots; one obtained by considering the expression +c positive, the other by considering this expression nega Clearing of fractions, 4x22x+13=10x2+5x NOTE. In completing the square, as the second term disappears when the root is extracted, we have written () in place. of it. 176. Whenever an equation has been reduced to the form x2+ bxc, its roots can be written at once; for this equation reduced (Art. 175) gives x=Hence, b The roots of an equation reduced to the form x2 + bx = c are equal to one half the coefficient of x with the opposite sign, plus or minus the square root of the sum of the square of one half this coefficient and the second member of the equation. In accordance with this, find the roots of x in the following equations: SECOND METHOD OF COMPLETING THE SQUARE. 177. The method already given for completing the square can be used in all cases; but it often leads to inconvenient fractions. The more difficult fractions are introduced by dividing the equation by the coefficient of x2, to reduce it to the form x2 + bx = c. To present a method of completing the square without introducing these fractions, we will reduce equation (1) in Art. 174. Multiplying (1) by a, the coefficient of x2, we obtain (2), in which the first term must be a perfect square. Since a dx, the second term, has in it as a factor the square root of a2x2, a2x2 can be the first term of the square of a binomial, and a dx the second term ; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the second term of the square divided by twice the square root of the first term of the square of the binomial, or a dx d and therefore the 2 ax 2' term required to complete the is square d2 4 which is the square of d2 one half of the coefficient of x in (1). Adding to both members 4 of (2), we obtain (3), whose first member is the square of a binomial. Extracting the square root of (3) and reducing, we obtain (5), or Hence, to reduce an affected quadratic equation, we have this second RULE. Reduce the equation to the form a x2 + d x = e; then multiply the equation by the coefficient of x2, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. |