ILLUSTRATIVE EXAMPLE. Solve the equation 3x2 8x=- - 4 To obtain for the equation the form, x2 + px = q, we divide by 3, and have 7x 3 (When the coefficient of x is negative, all the signs must be adding to both members the square of 7 6' (If the coefficient of x is a perfect square, it simplifies matters to merely add to both members the square of the quotient obtained by dividing the coefficient of the second term by twice the square root of the coefficient of the first,) as follows: 5 Adding the square of the result obtained by the above process, to both members, NOTE.-If the coefficient of x is not a perfect square, it may be made so by multiplication, i. e., multiplying all the terms. by the same number; as 18x2 + 5x = 2 multiplied by 2, gives, 36x2+10x = 4. Second method of completing the square. Reduce the equation to the form, ax2 + bx = c NOTE. The advantage of this method over the preceding is that fractions are avoided in completing the square. Multiply both members by 4 times the coefficient of x2, and add to each the square of the coefficient of x, in the given equation. Extract the square root of both members, and solve the simple equation thus formed. For example, multiplying each term of ax2 + bx = c by 4a, we have 4a2x2 + 4abx = 4ac Completing the square by adding square of b 4a2x2+4abx+ b2 = b2 + 4ac Extracting square root 2ax + b = ±√/b2 + 4ac Solve the equation 2x-7x= — 3 Adding to each member the square of 7, 16.2 56x+49= 24 + 49 = 25 Extracting square roots, This process may be shortened, if the coefficient of x in the given equation is an even number, fractions being avoided, and the rule modified as follows: Multiply both the members by the coefficient of x2, and add to each the square of half the coefficient of x. |