294. If the coefficient of x2 is 4, 9, 16, or any other perfect square, we may complete the square by adding to each side the square of the quotient obtained from dividing the second term by twice the square root of the first term. The square root of 4x2 is 2x, and 23x divided by twice 2x is 23 4 If the coefficient of x2 is not a perfect square, we may multiply the equation by a number that will make the coefficient of x2 a perfect square. Since the even root of a negative number is impossible, it is necessary to change the sign of each term. The resulting equation is 23. (x + 2) (2 x + 1) + (x − 1) (3 x + 2) = 57. 24. 3x (2 x + 5) − (x + 3) (3 x − 1) = 1. 25. (2x+5) (x-3) + x (3x+4) = 5. 26. (5 x2 - 8x - 6) – † (x2 − 3) = 2 x + 1. Another Method of Completing the Square. 295. If a complete quadratic is multiplied by four times the coefficient of x2, fractions will be avoided. The number added to complete the square by this last method is the square of the coefficient of x in the original equation 3x2 - 5x = 2. 296. If the coefficient of x is an even number, we may multiply by the coefficient of x2, and add to each member the square of half the coefficient of x in the given equation. Solve 3x2+4x= 20. Multiply by the coefficient of x2 and add to each side the square of half the coefficient of x, NOTE. If a trinomial is a perfect square, its root is found by taking the square root of the first and third terms and connecting these roots by the sign of the middle term. It is not necessary, therefore, in completing the square, to write the middle term, but its place may be indicated by a parenthesis, as in this example. Verify by putting the values of x in the given equation. x = 2. 3 (2)2 + 4 (2) = 20. 12+ 8 = 20. x = - 31. 3(-3)2+4(34) = 20. Solution by Resolving into Factors. 297. A quadratic which has been reduced to its simplest form, and has all its terms written on one side, may often have that side resolved by inspection into factors, and the roots found by putting each factor equal to zero. If either of the factors x + 12 or x-5 is 0, the product of the two factors is 0, and the equation is satisfied. Hence, the equation has three roots, 0, 3, — 2. (§ 130) x = 0, 3, or By the Factor Theorem (§ 135), we find that 1 put in place of x satisfies the equation, and is therefore a root of the equation. Divide by x 1, and resolve the quotient into its factors. (x-1) (x-4) (x + 3) = 0. Hence, the roots of the equation are 1, 4, — 3. 4. Solve x2 + 3 x - 10 = 0. the square of half the coefficient of x, to the first two 9 terms, we have a perfect trinomial square. Add and subtract ' |