Part 5

Affected. ax^2 + bx + c = 0.

Methods of solving Completing the square. Formula. Developed from ax^2 + bx + c = 0. Factoring.

Equations in the quadratic form.

Properties of quadratics r1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a). r2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a). Then r1 + r2 = -b/a. r1 . r2 = c/a. Discriminant, b^2 - 4ac, and its discussion. Nature or character of the roots.

~Simultaneous Quadratics~

CASE I.

One equation linear.

The other quadratic. 3x - y = 12, x^2 - y^2 = 16.

CASE II.

Both equations homogeneous and of the second degree. x^2 - xy + y^2 = 21, y^2 - 2xy = -15.

CASE III.

Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3, x^3 - y^3, x - y, x^2 +- xy + y^2, etc., given. x^2 + y^2 = 41, x + y = 9.

CASE IV. Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree. x^5 + y^5 = 242, x + y = 2.

CASE V. Special Devices

I. Solve for a compound unknown, like xy, x + y, (1)/(xy), etc., first. x^2y^2 + xy = 6, x + 2y = -5.

II. Divide the equations, member by member. x^4 - y^4 = 20, x^2 - y^2 = 5.

III. Eliminate the quadratic terms. 4x + 3y = 2xy, 7x - 5y = 5xy.

~Ratio and Proportion~

Proportionals mean, third, fourth.

Theorems 1. Product of means equals product of extremes. 2. If the product of two numbers equals the product of two other numbers, either pair, etc. 3. Alternation. 4. Inversion. 5. Composition. 6. Division. 7. Composition and division. 8. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent, etc.

Special method of proving four quantities in proportion. Let a/b = x, a = bx, etc.

~Progressions~

Development of formulas. { l = ar^(n - 1). { l = a + (n - 1)d. { S = (ar^n - a)/(r - 1). { S = (n/2)(a + l). { S = (rl - a)/(r - 1). { S = (n/2)[2a + (n - 1)d]. { S[infinity] = (a)/(1 - r).

Insertion of means Arithmetical. Geometrical.

~Binomial Theorem~

Review of binomial theorem laws. See Involution.

Expansion of (a + b)^n.

Finding any term by key number method. r^(th) or (r + 1)^(th) term method.