1. Deriving the Quadratic Formula

Properties of Algebra

Commutative Property of Algebra


  1. involving the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved, e.g. a × b = b × a.
  2. relating to or involving substitution or exchange.

Math block

a+b=7+2=9a + b = 7 + 2 = 9

Math block

b+a=2+7=9b + a = 2 + 7 = 9

Associativity Property of Algebra

Grouping of more than two numbers to perform basic aritmetic operations of addition/multiplication does not affect the final result.

Math block

(a+b)+c=(2+4)+5=7(a + b) + c = (-2 + 4) + 5 = 7

Math block

a+(b+c)=2+(4+5)=7a + (b + c) = -2 + (4 + 5) = 7

Distributive Property of Algebra

The distributive property defines that the product of a single term and a sum or difference of two or more terms inside the bracket is same as multiplying each addend by the single term and then adding or subtracting the products.

Math block

(a+b)c=ac+bc(a + b) \cdot c = a \cdot c + b \cdot c

Math block

a(b+c)=ab+aca \cdot (b + c) = a \cdot b + a \cdot c

Additive Identity Property

Math block

a=a+0a = a + 0

Multiplicative Identity Property

Math block

a=a1=1aa = a \cdot 1 = 1 \cdot a

Additive Inverse Property

Math block

a+(a)=0=(a)+aa + (-a) = 0 = (-a) + a

Multiplicative Inverse Property

Math block

212=0=1222 \cdot \frac{1}{2} = 0 = \frac{1}{2} \cdot 2

Logarithmic Properties

Product Rule

The log of a product is equal to the sum of the log of the first base and the log of the second base:

Math block

logb(xy)=logbx+logby\log_b (xy) = \log_b x + \log_b y

Quotient Rule

The log of a quotient is equal to the difference of the logs of the numerator and denominator:

Math block

logb(x/y)=logbxlogby\log_b (x/y) = \log_b x - \log_b y

Power Rule

The log of a power is equal to the power times the log of the base:

Math block

logb(xn)=nlogbx\log_b (x^n) = n \log_b x

Change of Base Formula

The log of a new base is the log of the new base divided by the log of the old base in the new base:

Math block

logbx=logax/logab\log_b x = \log_a x / \log_a b

Base Switch Rule

Math block

logbx=1logxb\log_b x = \frac{1}{\log_x b}