7. Find a quantity such that by adding it to each of the quantities a, b, c, d, we obtain four quantities in proportion. 8. What values must be given to a and b, so that (3a + 2b + 17)/2, (2a - 3b + 25)/3, 4 - 5a - 13b may be equal? ~MOUNT HOLYOKE COLLEGE~ ELEMENTARY ALGEBRA TIME: TWO HOURS 1. Factor the following expressions: (_a_) a^(3/4) - b^(3/4). (_b_) x^2 y^2 z^2 - x^2 z - y^2 z + 1. (_c_) 16(x + y)^4 - (2x - y)^4. 2. (_a_) Simplify (a^2 + b^2){(b^4)/(b^2 - a^2) - a^2}/{a/(a + b) + b/(a - b)}}. (_b_) Extract the square root of x^4 - 2x^3 + 5x^2 - 4x + 4. 3. Solve the following equations: (_a_) 1/x + 1/y = 5, 1/(x^2) + 1/(y^2) = 13. (_b_) x^2 - 5x + 2 = 0. (_c_) [27x + 1]^(1/2) = 2 - 3[3x^(1/2)]. 4. Simplify: (_a_) 7[54]^(1/3) + 256^(1/6) + [432/(-250)]^(1/3). (_b_) 1/[(a - b)(b - c)] + 1/[(c - a)(b - a)]. (_c_) Find [19 - 8[3^(1/2)]]^(1/2). 5. Plot the graphs of the following system, and determine the solution from the point of intersection: { x - 2y = 0, { 2x - 3y = 4. 6. (_a_) Derive the formula for the solution of ax^2 + bx + c = 0. (_b_) Determine the value of m for which the roots of 2x^2 + 4x + m = 0 are (i) equal, (ii) real, (iii) imaginary. (_c_) Form the quadratic equation whose roots are 2 + 3^(1/2) and 2 - 3^(1/2). 7. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches? 8. (_a_) In an arithmetical progression the sum of the first six terms is 261, and the sum of the first nine terms is 297. Find the common difference. (_b_) Three numbers whose sum is 27 are in arithmetical progression. If 1 is added to the first, 3 to the second, and 11 to the third, the sums will be in geometrical progression. Find the numbers. (_c_) Derive the formula for the sum of _n_ terms of a geometrical progression. 9. (_a_) Expand and simplify (2a^2 - 3x^3)^7. (_b_) For what value of x will the ratio 7 + x : 12 + x be equal to the ratio 5 : 6? ~UNIVERSITY OF PENNSYLVANIA~ ELEMENTARY ALGEBRA TIME: THREE HOURS 1. Simplify: [(a + x)/(a - x) - (a - x)/(a + x)] / (4ax)/(a^2 - x^2). 2. Find the H. C. F. and L. C. M. of 10ab^2(x^2 - 2ax), 15a^3b(x^2 - ax - 2a^2), 25b^3(x^2 - a^2)^2. 3. A grocer buys eggs at 4 for 7c. He sells 1/4 of them at 5 for 12c, and the rest at 6 for 11c, making 27c by the transaction. How many eggs does he buy? 4. Solve for t: (t + 4a + b)/(t + a + b) - (4t - a - 2b)/(t + a - b) = -3. 5. Find the square root of a^2 - (3/2)a^(3/2) - (3/2)a^(1/2) + (41/16)a + 1. 6. (_a_) For what values of m will the roots of 2x^2 + 3mx = -2 be equal?