Quadratic equations of the forms x2 = c, x2 + bx = 0, are called incomplete quadratic equations, because either the term involving x to the first power or the absolute term c is absent. 71. Incomplete quadratic equations are easily solved. Take the form ± √c. The form x2+ bx = 0 is solved by factoring, but may be solved also by the method of "completing the square," to be explained later. Factoring, we obtain, Make the first factor equal to zero, x(x + b) = 0, x = 0. 72. The solution of quadratic equations leads to two values of the unknown quantity. As both of these values are usually of interest and importance in the solution of problems, it is customary, in the extraction of square roots, to write down both results, the principal value and also the second value. This is indicated by the use of the symbol ±. Thus, in "x= ± √c," +Ve is the principal root, Ve is the second root. Since, in finding =Vc, the square root of both sides of the equation has been extracted, it might be claimed that the sign should be written on both sides, giving But this result is the same as when we write x = Of these four sets, the first two are the same, and the last two are the same. Hence, x = Ve gives all the values of x. E 73. A complete quadratic equation may be solved in three ways: (1) By factoring, (2) By completing the square, (3) By substitution in a formula obtained by the method of completing the square. I. With our present knowledge of factoring, the first method is applicable only when the roots of the quadratic equation are rational. The method depends upon the principle that, if the product of two or more factors equals zero, one factor must equal zero. Transpose all the terms to the first side, 3 x2 - 5 x − 2 = 0. Place the first factor equal to zero, (3x+1)(x − 2) = 0. 3x+1=0. II. The second method depends on the type form of a trinomial which is a perfect square: a2±2 ab + b2. If the b2 is lacking, we may take the square root of the first term, double it, divide the middle term by this, and square the quotient. Take, for illustration, x2 ± 5 x. To complete the square, we take the principal square root of x2, which is x; double it, 2x; divide 5 x by 2x, ; square, 25. Hence x2+5x+25 is a perfect square. This results in the following rule for the solution of a quadratic equation: 1. Transpose all terms containing x2 and x to the left side of the equation; all others to the right side. 2. Divide both sides by the coefficient of x2. 3. Add to both sides the square of half the coefficient of x. 4. Extract the square root of both sides and solve the resulting equations. III. The third method depends upon the formula derived from the solution of the type form of the complete quadratic. In using the formula, transpose all terms of the given equation to the left side. Why? 3x2-5x= 2. 3x2-5 x 2 = 0. 5, c = - -2. 5+√25 +24 5 ± 7 = = 2 or -. 6 EXERCISES 74. Solve; if a numerical equation has irrational roots, approximate their values, to three decimal places. Use the table of square roots in § 197. 1. x2+4x= 5. 2. 3x2+x-14=0. 3. x28x=-11. 4. x2-2x-15= 0. 16. (x+2)(x-2)=7(x+2)-5. 17. (x+3)-2(x+3)+ 1 = 0. 18. 5 x(x-3)- 2(x2 − 6) = (x +3)(x+4). x+2 x 5 2x+1 19. 1 20. = x 3 n2 Ascertain in each problem whether both roots of the quadratic equation are applicable to the problem. 75. 1. Find two consecutive numbers whose product is 992. 2. Find the length and breadth of a rectangle whose area is 375 sq. in., and whose length exceeds its breadth by 10 in. 3. Express 71 as the sum of two numbers whose product is 448. 4. If the length of a square be increased by 2, and the width be increased by 3, the area of the resulting rectangle is 40. Determine the length of a side of the square. 5. The height of a triangle exceeds its base by 8; if the area of the triangle is 1209, what is its base? 6. The diagonal of a square is 2 ft. longer than the side. Find the side. 7. A cylinder 12 ft. in height has a capacity 125 cu. ft. Determine the diameter of its base. 8. When the edges of a cube are each increased by 6 in., the volume is increased by 936 cu. in. Find the dimensions of the original cube. 9. The sum of the numerator and denominator of a fraction is 77. If the numerator is increased by 111 and the denominator is increased by 40, the fraction is doubled. Find the fraction. 10. Find two numbers which differ by 2, the cubes of which differ by 296. 11. The radius of one circle is twice the radius of another. Find the radii of both, if the difference of their areas is 75. 12. The difference of the volumes of two in.; the difference of their radii is 5 in. correct to two decimal places. spheres is 100 cu. Find their radii, 13. A woman paid $ 64 for silk. If she had bought 4 yards less for the same money, she would have paid $13 more per yard. How many yards did she buy? 14. The longer leg of a right triangle exceeds the shorter leg by 3 ft. The area of the triangle is 135 sq. ft. Find the length of each leg. 15. A bookdealer sells a number of algebras for $87. Had he reduced the price of each book by 12 ¢, he would have sold 16 more books for the same sum of money. How many books would he have sold at the reduced rate? |