Algebra I Basics Flashcards
what is the quadratic equation?
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.
what is the 0-product property
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.
factor:
3x2+6x+9=0
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.
what is foil?
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.
factoring trinomials
a is not 1
- 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).
Restricted values
any value in a rational expression that could possibly make the denominator = 0.
Constant of proportionality
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 xa, too.
Inverse variation
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…..
distance formula
d = sqrt((x1-x2)2+(y1-y2)2)
pythagorian theroem for tryhards(losers)
Fractional Exponents
where 1/n
x1/n
basically the nth root of any number: n√x
Fractional Exponents
where m/n
xm/n
turns into:
n√xm
where n is the radical and m is an exponent.
Square Root Property
Quadratic Formula by Completing the Square
- 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
Quadratic formula
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.
Proportional Equation
(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
