Algebra I Basics

Question 1

what is the quadratic equation?

Answer 1

an equation that contains a squared variable and no other term with a higher degree.

ax²+bx+c = 0; ax² = 0; e.t.c…

since “a” is being multiplied by a squared variable, it is a real number.

Question 2

what is the 0-product property

Answer 2

a rule that declares “if the product of two numbers is zero, then at lest one of them must also be 0.”

i.e. xy=0
x could be 0.
y could be 0.

Question 3

factor:

3x²+6x+9=0

Answer 3

3(x²+2x+3)

we get here by factoring all terms by their biggest component, 3. At this point we end up with the aforementioned equation. However, to further factor this equation we’d need to factor x²+2x+3, which is unable to be factored with out the use of complex numbers. No calc for today.

Question 4

what is foil?

Answer 4

first, outer, inner, last: the best way to teach a toddler to multiply a binomial.

i.e. (2+x) (3-x)
2 * 3 = 6
2 * -x = -2x
x * 3 = 3x
x * -x = -x²
now add like terms.
-x²+x+6
now you have quadratic equation. cry about it.

Question 5

factoring trinomials

a is not 1

Answer 5

• factor out any common factor
• look for integers whose product is ac and sum is b
• use the integers to rewrite the middle term to be a four-term polynomial
• factor the polynomial.

youre not allowed to factor trinomials, dummy!

example problem:
12x²+27x+6
3(4x²+9x+2)
a = 4, b =9, c = 2
a * c = 8. Integers with products of 8 and sums of 9 can easily be found through factoring the product of ac.
8 + 1 fits the situation.
rewrite.
3(4x²+8x+1x+2). 8x+1x = 9x, so this is legal in the problem.
factor.
3(4x(x+2)+1(x+2))
3(x+2)(4x+1).

Question 6

Restricted values

Answer 6

any value in a rational expression that could possibly make the denominator = 0.

Question 7

Constant of proportionality

Answer 7

y=kx
k represents the change in y per one unit of x

you find k by doing k= y/x.
this can be done where x_a, too.

Question 8

Inverse variation

Answer 8

y = k/x
where y varies inversely to x.

if y goes up, x goes down. vise versa.

found by writing k = xy, solving for k, and doing y = k/x.
so: if y=2 x=20. where we find y when x=73
k= 2(20) = 40
y = 40/73
y = .54…..

Question 9

distance formula

Answer 9

d = sqrt((x₁-x₂)²+(y₁-y₂)²)

pythagorian theroem for tryhards(losers)

Question 10

Fractional Exponents

where 1/n

Answer 10

x^1/n
basically the nth root of any number: ⁿ√x

Question 11

Fractional Exponents

where m/n

Answer 11

x^m/n
turns into:
ⁿ√x^m

where n is the radical and m is an exponent.

Question 12

Square Root Property

Answer 12

±√x

This is a factoring rule, wherein, you can quite easily just take an equation like x²=144, and just apply ±√144, and now you have ±12, which, when -12(-12)=144, and when 12(12)=144, makes sense.

Question 13

Quadratic Formula by Completing the Square

Answer 13

• Write the equation where ax² and bx are on one side, and the constant on the other.
• Make sure the coefficient “a” is 1, if not, divide both sides by coefficient.
• Complete the square by multiplying one-half squared of the coefficient “b” both sides of the equation.
• Factor remaining trinomial into binomial, using “factoring trinomials”
• Cancel duplicate
• Simplify into (ax+b)²=c
• Solve the equation using the square root property

i.e.
4x²+40x+8 = 0
4x²+40x = -8
4x²+40x = -8 <—— divide all by 4
x²+10x = -2 <—— (10/2)²
x²+10x+25= -2+25
x²+10x+25=23<——- factor
find factors of ac {a=1,b=10,c=25}
ac = 25, factors of 25 are {1,5,25}
5+5<— add factors to get b
rewrite to integrate the new numbers
x²+5x+5x+25=23 <—— now factor into two binomials of (a+bx₁)+(bx₂+25)=23
when factored, x(x+5)+5(x+5)=23 <—- follow trinomial factoring, and cancel second term
(x+5)(x+5)=23 <—– now simplify
(x+5)²=23
now follow square root, √(x+5)²=√23
x+5≈±4.796 <——- two solutions.
(x+(5-5)≈4.796-5, x = -2.06)
(x+(5-5)≈ -4.796-5, x= -9.796)
so the double root is, x = -9.796, -0.206

Question 14

Quadratic formula

Answer 14

x = (-b ± √b²-4ac) / 2a

you can basically determine this whole thing based on the discriminant: b² - 4ac. if b² - 4ac = 0 there is one solution. if b² - 4ac > 0, two solutions. if b² - 4ac < 0, complex solution.

Question 15

Proportional Equation

Answer 15

(n/m)=(x/y)

in this equation, you’re solving for one of the variables and all others are given

to solve:
n(y)=m(x)
ny=mx
ny/m=mx/m
ny/m=x <———- solved

Question 16

Fraction Equations

Answer 16

a/b ± c/d(ex)=±f/gx ± h/i

where variables except x represent numbers

to solve:
3x-5/6 - 1/8= 5/8x + 3/4(x+3) - 7/12 <—multiply by least common multiple
(3x-5/6 - 1/8= 5/8x + 3/4(x+3) - 7/12)24 <—ignore brackets and things in numerator
4(3x-5)-3=15x+18(x+3)-14 <—solve
-21 = x

Question 17

Literal Equations

Answer 17

no numbers only variables

Question 18

Simplifying Radicals

Answer 18

• Square Roots, Cube Roots and x Roots don’t require simplification.
• Irrational roots do require simplifaction. Shown below
• Irrational roots with coefficients require simplifaction.

Irrational root simplification:
Find factors of the radicand that containa perfect square.
√500 has factors of 25 and 20, 25 being a perfect square. Take the square of 25, being 5 and make it a coefficient to the radical.
the final equation should look like 5√20.
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Irrational root simplification with coefficients simplification:
4√18 <———— factor radicand
4√9·2 <———— take out the perfect square, square it, and multiply it by the coeffient
4·3√2 <————– multiply coeffients
12√2 is the final equation

Question 19

Conversion

Answer 19

take equivalent factors and convert to different units

For example:
150mph to meters per minute
150mile/1hour · 5280ft/1mile <——- cancel miles
150 · 5280ft/1hour <——- multiply
792000ft/1hour · 0.3048meters/1foot<— cancel feet
792000x0.3048meters/1hour <——- multiply
241401.6meters/1hour · 1hour/60minutes <– cancel hours
241201.6meters/60minutes <—divide
4023.36 meters per minute
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rembember you can still cross multiply even if the equation is flipped and your like terms are in denominator and numerator.

Question 20

ℝ

Answer 20

A domain representing the whole of all real numbers.

Question 21

Solving a Linear System in Three Variables

Answer 21

• Write each equation in the format of: ax + by + cz = d
• Move all variable terms to the left and the constant term to the right
• Pick one of the variables to eliminate
• Use elimination to eliminate one of the variables from any two equations
• Use elimination again to eliminate the same variable from any other two equations
• Solve the linear system in two variables
• Find the final unknown
• Plug in the values for the two known variables into any original equation, solve for the final unknown

Question 22

Matrix

the matrix

Answer 22

A way of grouping constants
shown like this:
[1 -7 -6]
[-9 2 -4]