XXI. QUADRATIC EQUATIONS. 258. A Quadratic Equation, or an equation of the second degree (Art. 167), is one in which the square is the highest power of the unknown quantity. A Pure Quadratic Equation is one which contains only the square of the unknown quantity; as, ax2 = b. An Affected Quadratic Equation is one which contains both the square and the first power of the unknown quantity; as, ax2 + bx + c = 0. PURE QUADRATIC EQUATIONS. 259. A pure quadratic equation is solved by reducing it to the form x = a, and then extracting the square roots of both members. 1. Solve the equation 3x2+ 7 = 5 x2 4 +35. Clearing of fractions, 12x2 + 28 = 5 x2 + 140. Note 1. The double sign is placed before the result because the square root of a number is either positive or negative (Art. 201). Note 2. Since the square root of a negative quantity is imaginary (Art. 246), the values of x can only be indicated. 10. (3x - 2) (2x+5) + (5x + 1 ) (4x - 3) — 91 = 0. 15. (2x - a) (x - b) + (2x + a) (x+b)= a2 + b2. 16. 51 - 2 - 3892 = 2. 5x2-1 3x2+1 x2-3 x2+2 (x2-3) (x2+2) AFFECTED QUADRATIC EQUATIONS. 260. An affected quadratic equation is solved by adding to both members such a quantity as will make the first member a perfect square; an operation which is termed completing the square. FIRST METHOD OF COMPLETING THE SQUARE. 261. Every affected quadratic equation can be reduced to the form x2+px=q; where p and q represent any quantities whatever, positive or negative, integral or fractional. Let it be required to solve the equation + 3x = 4. In any trinomial square (Art. 108), the middle term is twice the product of the square roots of the first and third terms; hence the square root of the third term is equal to the second term divided by twice the square root of the first. Therefore the square root of the quantity which must be 3 x 3 added to x2+3x to make it a perfect square, is 一, or 2x 2 3 9 or , we have 4 Adding to both members the square of 262. From the above operation we derive the following rule: Reduce the equation to the form x2+ px = q. Complete the square by adding to both members the square of half the coefficient of x. Extract the square root of both members, and solve the simple equation thus formed. Note. These values may be verified as follows: Putting x = 2 in the given equation, 12-16 = 4. 2 Ans. নলি 3' If the coefficient of 2 is negative, it is necessary to change the sign of each term. Whence, x=-=-or-, Ans. EXAMPLES. Solve the following equations : 3. x2 + 4x = 5. 4. x2- 5x = -4. 5. x2-7x = - 12. 6. x2+x=6. 7. 3x2-4x = 4. 8.2x2+5x = -2. 10. 4x2-3 = 11 x. 12. 14+15 x - 9x2 = 0. 263. If the coefficient of 2 is a perfect square, it is convenient to complete the square directly by the principle of Art. 261; that is, by adding to both members the square of the quotient obtained by dividing the second term by twice the square root of the first. 1. Solve the equation 9x2 - 5 x = 4. The quotient of the second term divided by twice the square root of the first, is. Adding the square of both members, 6 5 6 to Note. If the coefficient of x is not a perfect square, it may be made so by multiplication. Thus, in the equation 18 x2 + 5 x = 2, the coefficient of x2 may be made a perfect square by multiplying each term by 2. If the coefficient of x2 is negative, the sign of each term must be changed. |