12. How many gallons each of cream containing 33% butter fat and milk containing 6% butter fat must be mixed to produce 10 gallons of cream containing 25% butter fat? 13. I have $6 in dimes, quarters, and half-dollars, there being 33 coins in all. The number of dimes and quarters together is ten times the number of half-dollars. How many coins of each kind are there? (_College Entrance Board._) ~Reference:~ The last part of the chapter on Quadratic Equations in any algebra. THE THEORY OF QUADRATIC EQUATIONS ~I. To find the sum and the product of the roots.~ The general quadratic equation is ax^2 + bx + c = 0. (1) Or, x^2 + (b/a)x + c/a = 0. (2) To derive the formula, we have by transposing x^2 + (b/a)x = -c/a. Completing the square, x^2 + (b/a)x + [b/2a]^2 = (b^2)/(4a^2) - c/a = (b^2 - 4ac)/(4a^2). Extracting square root, x + b/2a = [+-[b^2 - 4ac]^(1/2)]/(2a). Transposing, x = -b/2a +- [[b^2 - 4ac]^(1/2)]/(2a). Hence, x = [-b +- [b^2 - 4ac]^(1/2)]/(2a). These two values of x we call _roots_. For convenience represent them by r_1 and r_2. Hence, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). --------------------------------------------- Adding, r_1 + r_2 = -(2b)/(2a) = -b/a. (3) Also, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). ------------------------------------------- Multiplying, r_1 r_2 = (b^2)/(4a^2) - (b^2 - 4ac)/(4a^2) = (b^2 - b^2 + 4ac)/(4a^2) = (4ac)/(4a^2) = c/a. (4) Hence we have shown that r_1 + r_2 = -b/a, and r_1 r_2 = c/a. Or, referring to equation (2) above, we have the following rule: _When the coefficient of x^2 is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term._ EXAMPLES: 1. x^2 - 9x + 21 = 0. Sum of the roots = 9. Products of the roots = 21. 2. 3x^2 - 7x - 18 = 0. Sum of the roots = 7/3. Product of the roots = -6. 3. -21x = 17 - 4x^2. Sum of the roots = 21/4. Product of the roots = -17/4. ~II. To find the nature or character of the roots.~ As before, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a), r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). The [b^2 - 4ac]^(1/2) determines the _nature_ or _character_ of the roots; hence it is called the _discriminant_. ~If b^2 - 4ac is positive, the roots are real, unequal, and either rational or irrational.~ ~If b^2 - 4ac is negative, the roots are imaginary and unequal.~ ~If b^2 - 4ac is zero, the roots are real, equal, and rational.~ EXAMPLES: 1. x^2 - 4x + 2 = 0.