2. Solve 3x+2x=33. Solution Divide by 3, 2 Add (+2√T)', or ()', 2 + 3 + = 11+ = 100 Extract √, Transpose, 9 2 1 1 x 9 9 (A) (1) (3) Therefore, Scholium 1.-When the coefficient of x is 1, the quantity to be added to both members to complete the square is the square of half the coefficient of x. Scholium 2.-When the equation is multiplied through by the coefficient of x2, the quantity to be added to both members is the square of half the coefficient of x in the typical equation. This method is generally the best, as it avoids all fractions above and below fourths. 4. Solve 2x2 + 3 x = 14. Solution: Multiply by 4 × 2, 16 x2 + 24 x = 112 1 - 3 Therefore, Scholium 3.-When the equation is multiplied through by four times the coefficient of x2, the quantity to be added to both members is the square of the coefficient of x in the typical equation. This is called the Hindoo method of completing the square. It avoids all fractions, but often gives rise to very large whole numbers. 3. Solution of Literal Affected Quadratics. Illustrations.-1. Solve x2+ax=b. Solution: Complete the square, 210. When an equation contains two and only two exponents of the unknown terms, and one of them is twice the other, it is said to have the quadratic form; as, x+6x=16, ax + bx3 = c, or (a + b x) + p (a + b x)3 = c. 211. Any equation having the quadratic form, whatever its degree, may be solved by any of the methods employed to solve an affected quadratic. Illustrations.-1. Solve x + 6 x2 = 16. Solution: Complete the square, (A) Extract √, Transpose, Extract √, x= ±√2, or 2√2 |