610. Extract the square root of both members of the following equations, adding to both, where necessary, such a number as will make the first member a complete square. 1. x2+6x+9=40+ 9. 2. x2-12x+36=28+36. Remember that (+7) × (+7) = 49, and that (–7) × ( − 7) = 49. ... √49 +7 or 7, written ± 7. = - Completing the square, we have x2 - 10x+25= 24 + 25 = 49. Extracting the square root of both sides, we have the = 56. 14. x2+22 x = 104. 15. x2 16x=- 64. 16. x2+36 x = 76. To make the first member a complete square, you added square of what part of the coefficient of x? If one of two factors is zero, the product is zero. The converse is also true. If the product of two factors is zero, one of the factors is zero. Given (x2)(x-3)= 0. One of the factors in the above equation is equal to zero. 617. A quadratic equation may sometimes be readily solved by factoring. 1. The sum of two numbers is 12; their product is 32. What are the numbers? x and 12 x = numbers. (12 - x) x = product. 2. The base of a rectangle is 50 feet longer than its altitude. x Its area is 2400 square feet. How long is the base? Area x 250 x 2400 sq. ft. x + 50 3. The perpendicular of a right-angled triangle measures 15 yards more than the base. The hypotenuse is 75 yards. Find the length of the perpendicular. x2 + (15+ x)2 = 752. The 4. The hypotenuse of a right-angled triangle is 14 times as long as the base. area of the triangle is 150 square yards. long is the hypotenuse? How Perpendicular = √(x)2x2; area = base × per 5. The entire surface of a square prism is 170 square feet. Its altitude is 6 feet, and one side of its base is x feet. Find the value of x. 6. A garden 50 feet long, 40 feet wide, has a walk just outside it x feet wide. Find the area of the walk. If the area of the walk is 784 square feet, what is its width? 7. A field, ABCD, contains 12 acres. Its length is 17 times its breadth. How many rods long is the diagonal BC? 40+ 2 x 40 50+ 2 x 50 21 ريع 15 X A rods 8 B 8. A flag-staff, AB, 50 feet high, was broken off at the point C. The broken part, resting on C, reached the ground D, 30 feet from the base of the staff. Find the length of the part broken off. 15+x 9. A ladder, CE or DE, placed at a point E, in a street 58 feet wide between the opposite houses, just touches the top of a house, DB, 60 feet high on one side of the street, or the top of a house, CA, 56 feet high on the other side. Find the length of the ladder. DE2 = 602+(58 — x)2= CE2 = 562 + 2a2. 10. ABC is a triangle. The side AB measures 13 feet; the side BC, 4 feet; AC, 15 feet. Find the altitude BD. 58-x E Br BD2 = AB2 - AD2 = BC2 — CD3. 11. ABCD is a trapezium. AB= 34 feet; BC = 20 feet; CD = 40 feet; DA 26 feet. The perpendicular BF measures 16 feet. Find the length of the diagonal AC and of the perpendicular ED. A E B |