Part 21

~Reference:~ The chapter on Quadratic Equations in any algebra (first part of the chapter).

1. Solve by three methods--formula, factoring, and completing the square: x^2 + 10x = 24.

Review equations in the quadratic form and solve:

2. x^4 - 5x^2 = -4.

3. 2[x^(-2)]^(1/3) - 3[x^(-1)]^(1/3) = 2.

4. (x + 3)/(x - 3) + 6 = 5[(x + 3)/(x - 3)]^(1/2). (Let y = [(x + 3)/(x - 3)]^(1/2) and substitute.)

5. 3x^2 - 4x + 2[3x^2 - 4x - 6]^(1/2) = 21.

6. x^2 + 5x - 5 = (6)/(x^2 + 5x).

Solve and check:

7. [x + 7]^(1/2) + [3x - 2]^(1/2) = (4x + 9)/([3x - 2]^(1/2)).

8. [x^2 - 5]^(1/2) + 6/[[x^2 - 5]^(1/2)] = 5.

9. (10w)/([10w - 9]^(1/2)) - [10w + 2]^(1/2) = 2/([10w - 9]^(1/2)).

Give results by inspection:

10. (a^(1/2) + b^(1/2))(a^(1/2) - b^(1/2)).

11. ([10 + 19^(1/2)]^(1/2))([10 - 19^(1/2)]^(1/2)).

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).