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+ bx=c, its roots can be written at once; for

this equation reduced (Art. 175) gives xHence,

b

± +c.

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:

1. Reduce 2+8x=65.

x=4±√16655, or 13, Ans.

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 ad x, 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

term required to complete the square

one half of the coefficient of x in (1).

ad x

2 ax

=

d

; and therefore the

d2

is

4

which is the square of

d2

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+ dx=e; then mul tiply 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.

NOTE 1. This method does not introduce fractions into the equation when the numerical part of the coefficient of x is even. When the coefficient of x is unity, this method becomes the same as the first method.

NOTE 2. If the coefficient of x is already a perfect square the square can be completed without multiplying the equation, by adding to both members the square of the quotient arising from dividing the second term by twice the square root of the first. This method also becomes the same as the first method when the coefficient of is unity. NOTE 3. As an even root of a negative quantity is impossible or imaginary, the sign of the first term, if it is not positive, must be made so by changing the signs of all the terms of the equation.

Completing sq. by Note 2, 252-()+1=1+ 195 196

THIRD METHOD OF COMPLETING THE SQUARE.

178. The method of the preceding Article introduces fractions whenever the numerical coefficient of x is not even. To present a method of completing the square without introducing any fraction, we will again reduce equation (1) in Art. 174.

4a2x2+4 a d x + ď2 = ď2 + 4 a e

2 axd± √ ď2 + 4 ae

4

e

α

(3)

(4)

(5)

Dividing (1) by a, the coefficient of x2, we have (2); then completing the square according to the Rule in Art. 175, we have (3); and if we multiply (3) by 4 a3, it will give (4), an equation free from fractions (unless a, d, or e in (1) are themselves fractions), and one whose first member is the square of a binomial. To produce this equation directly from (1), we have only to multiply (1) by 4a; i. e. by four times the coefficient of x2, and add to both members ď2; i. e. the square of the coefficient of x. Reducing we have (6), which is a general expression for the value of x in any equation in the form of ax2+dx=e.