Algebra 3 Flashcards
Surd equations
Get the √ on its own on 1 side of the equation.
Square both sides => removing the √
Solve the resulting equation.
Why do you check both solutions of the original surd equation
When squaring both sides an EXTRA root can be generated which does not satisfy the original equation.
√9 =
3 not +/-3
Natural Numbers
POSITIVE WHOLE numbers →N = {𝟏,𝟐,𝟑,𝟒…} DOTS.
Integers
ANY WHOLE number, POSITIVE, NEGATIVE or ZERO →Z = {…−𝟑,−𝟐,−𝟏,𝟎,𝟏,𝟐…} DOTS.
Real numbers
ANY number that can be PLOTTED on a Number line. THICK LINE.
When does the inequality flip
when you multiply or divide by a negative number
The lengths of any two sides in a triangle added together are always…
greater than the length of the third side.
Quadratic inequalities
Temporarily rewrite the inequality as an equation.
Solve this quadratic equation.
Use a rough sketch to determine the range of values of 𝒙.
> means ___ the X axis + the solution set will be __ inequalities ___ your values
> means ABOVE the x-axis and the solution set will be TWO inequalities EITHER SIDE of your values.
< means ___ the X axis + the solution set will be __ inequalities ___ you values
<means BELOW the x-axis and the solution set will be ONE inequality BETWEEN your values.
Rational inequalities
Multiply both sides by the (𝑫𝑬𝑵𝑶𝑴𝑰𝑵𝑨𝑻𝑶𝑹)² to ensure we’re multiply by something positive => don’t have to flip inequality
|x| is
the ABSOLUTE VALUE or MODULUS of x
The modulus is…
Geometrically it is how far a number is from zero on a number line.
ALWAYS a POSITIVE value
|x| = 3
x = 3 OR x=-3
To solve a modulus equation
Get the modulus on its own on one side of the equation.
SQUARE BOTH SIDES => removing the modulus.
Solve the resulting equation.
g(x) = 4
The line
Is parralel to X axis
Goes through 4 on the Y axis
If there are 2 modulus
get ONE ON EACH SIDE before squaring
Does squaring both sides work on modulus inequalities
Yes
Inequalities: Proofs
Get each term to 1 side and 0 on the other
(any real no.)²
≥ 0
-(any real no.)²
≤0
Discriminants
𝒂𝒙²+𝒃𝒙+𝒄=𝟎 is a quadratic equation whose roots are given by 𝒙=(−𝒃±√𝒃² −𝟒𝒂𝒄)-2a
The value of 𝒃² −𝟒𝒂𝒄 =DISCRIMINANT
It tells us if the ROOTS are EQUAL, REAL or NON REAL.
Real roots
𝒃² −𝟒𝒂𝒄 ≥ 0
Cuts x axis at 2 points
Equal roots
𝒃² −𝟒𝒂𝒄 = 0
cuts x axis at 1 point
Non real roots
𝒃² −𝟒𝒂𝒄 < 0
doesnt cut the x axis
a > 0 graph shape
U shape
a < 0 graph shape
n shape
Do you have to state the discriminant formula every time
YES
