Linear And Quadratic Equations Flashcards
Zero product property
If two things multiplied together equal zero then at least one of them is zero
Difference of squares
x^2-y^2=(x-y)(x+y)
What is the factored form of:
(x^2)(y^2)-16?
(xy-4)(xy+4)
Difference of squares ^
If x≠y, then (x-y)/(y-x)=?
-1
If two equations describe the same line, how many solutions does that system of equations have?
(Like 10x+10y=20, and x+y=2)
Infinitely many
what is the Least Common Multiple (LCM) of a set of numbers?
the smallest non-zero whole number that is divisible by all the numbers in the set
What is the quadratic expression of this common identity: (x+y)^2
x^2+2xy+y^2
What is the quadratic expression of this common identity: (x-y)^2
x^2-2xy+y^2
What is the quadratic expression of this common identity: (x-y)(x+y)
*what is this referred to as?
x^2-y^2
if a system of two linear equations are not equal to each other, how many solutions does the system have?
1
when can you divide by a variable
only when you know that variable is not equal to zero
what is the quadratic formula and why would you use it?
formula in pic - use it to find the roots/solutions of a quadratic equation
what form does a quadratic equation have to be in to utilize the quadratic formula to solve for its roots
general form: ax^2+bx+c = 0
when should you use the quadratic equation vs factoring?
when the expression cannot easily be factored
how can you use the discriminant in the quadratic equation to quickly determine the number of roots the equation has (and what is the discriminant)?
discriminant is the portion in the numerator under the radical - the b^2-4ac part. if the discriminant is positive, there are two roots, if it is zero there is one root, if it is negative there are zero roots (no solution). this makes sense since if it was zero, the +/- sqrt(discriminant) would yield the same thing for both +/-
what do vieta’s equations say are the sum and products of the solutions to equations (in general form) that have two solutions
sum of the two solutions is -b/a
product is c/a
hidden quadratics often contain a variable (x) with no exponent and then the square root of that same variable - what does that look like and how do you get it into quadratic form?
- look at attached picture
x-8*sqrt(x)+15=0
step 1: replace x with (sqrt(x))^2
step 2: replace sqrt(x) with y
step 3: solve resulting quadratic for y
step 4: replace y with sqrt(x) and solve for x
hidden difference of squares in: squared algebraic expressions
how would you use difference of squares to simplify (2x+1)^2 - (2x-4)^2
(2x+1)^2 - (2x-4)^2 = (2x+1-2x+4)(2x+1+2x-4) = (5)(4x-3) = 20x-15
hidden difference of squares in: implied difference of squares
can be helpful in finding the product of two numbers if one of the numbers is k more than some number n and the other is k less than that same number n. ex: 10,004*9,996
10,0049,996 = (10,000+4)(10,000-4) <-difference of squares
= 10,000^2 - 4^2 = 99,999,984
hidden difference of squares with roots:
must be familiar with property -> for x>=0, (sqrt(x))^2 = x
look at screenshot pic for work through example
