Linear and Quadratic Equations

Question 1

What is a linear equation?

Answer 1

A linear equation is an equation with one or more variables, in which each is raised only to the first power and those variables are not multiplied by one another.

e.g.,
10x + 20 = 30
or
5y - 6x = 14

“Raised to the first power” means we don’t often see the exponent.

Question 2

How do we maintain equality when solving linear equations?

How can we confirm we got the correct answer?

Answer 2

To maintain equality, whatever we do to one side of the equation, we must do to the other.

To check our work, we can plug in our answer and confirm whether both sides are equal.

Question 3

What are our two options for solving two variable equations?

Answer 3

Substitution method: Isolate one variable of one equation, then substitute it for that variable in the other equation. We then get a value, which we can use to solve for the other unknown variable.

Combination method: Stack the two equations, then add or subtract the entirety of one equation to (or from) the other.

Note: you can multiply equations by (-1) to make addition or subtraction easier.

Question 4

What is a coefficient?

Answer 4

The number a variable is multiplied by in an equation.

e.g.,
2x + 5y = 21 (The coefficient of x is 2; the coefficient of y is 5).
or
x + 3 = 4 (The coefficient of x is 1).

Question 5

How do we approach systems of equations when our coefficients don’t line up?

e.g.,
(1) 4x + 3y = 12
(2) 3x + 2y = 10

Answer 5

We can manipulate one or both of them to make the variables cancel.

e.g., we can multiply (1) by 3 and (2) by 4 to get 12x for both, then cancel through subtraction as normal.

Question 6

When do we use the substitution method vs. the combination method?

e.g.,
(1) 2x + 4y = 25 and 14x + 7y = 100
(2) 3y = 12 + 6x and 15y + 20x = 5.5
(3) 55x + 10y = 101 and 2x + y = 76

Answer 6

Use the substitution method when you can easily isolate one of the variables.

Use the combination method when you cannot easily isolate one of the variables.

e.g.,
(1) Combination
(2) Substitution
(3) Combination

Question 7

True or False: we can only substitute single variables, not expressions.

If true, what type of problem is this most common with?

e.g.,
(1) x + y = 12
(2) x + y + 3z = 14

or
(1) 18a + 39b = 21
(2) c + 6a + 13b = 20

Answer 7

False. We can substitute entire expressions into equations.

This is most pertinent when we have two equations, but three variables.

e.g.,
(1) x + y = 12
(2) x + y + 3z = 14 therefore (12) + 3z = 14

and

(1) 18a + 39b = 21
(2) c + 6a + 13b = 20, so c + (7) = 20

Question 8

What is our approach for problems with fractions?

e.g., (2x/3) - 2y = (5/2)

or

(x/5) - (2y/x) = 3

Answer 8

We multiply both sides by the Least Common Denominator to get rid of the fractions, then solve as we would a normal problem.

e.g., (2x/3) - 2y = (5/2) would be multiplied by (6/1)

or

(x/5) - (2y/x) = 3 would be multiplied by (5x/1).

Question 9

How do we eliminate a fraction if it is not yet distributed?

e.g., x - 5 = (2/3)(y + 10), how would we eliminate the fraction?

Answer 9

We can distribute the (2/3), then eliminate, or we can eliminate it immediately.

If eliminating before distributing, we do not need to multiply (y+10) by the LCD.

e.g., x - 5 = (2/3)(y + 10), we multiply ( x - 5 ) and (2/3) by the LCD of 3, but not the ( y + 10 ).

See example 15.

Question 10

What does it mean when we’re asked “what is (x) in terms of (y)?”

How about when we’re asked to “solve for (a) in terms of (b) and (c)?”

Answer 10

What is (x) in terms of (y) means we isolate x and set it equal to some expression of y. So y is the only variable in the answer.

“Solve for (a) in terms of (b) and (c)” means we isolate (a) and set it equal to some expression of (b) and (c), so (b) and (c) are the only variables in the answer.

Question 11

Note: when all terms in an equation or expression share a common term, you can factor the common factor out.

Answer 11

e.g., 4x + 4y = 16
4 (x + y) = 4(4)

4x + 4y = 19
4 (x+y) = 19

Question 12

If the product of two integers is 1, what does this indicate?

Answer 12

Either both integers are equal to 1 or both are equal to -1.

Question 13

What does the Zero Product Property tell us?

Answer 13

If the product of two quantities is 0, one or both of those quantities must be equal to 0.

e.g., If a x b = 0, a or b or both must equal 0.

If (n + 4)(n - 1) = 0, either (n + 4) = 0 or (n - 1) = 0.

Question 14

In an equation like x(x + 100) = 0, what is a common mistake that we want to avoid?

Answer 14

Don’t divide both sides by x to remove the x outside the parenthesis! YOU CAN’T DIVIDE BY A VARIABLE, AS IT COULD BE 0.

There are two possible solutions, x = 0 and x + 100 = 0.

The mistake is to divide by x, leaving us with one potential solution, x = -100.

Recall the Zero Product Property: if two things multiplied equal 0, at least one of those things must be zero.

Question 15

*****Can we divide by a variable?

e.g., x(x + 100) = 0

Answer 15

() Confirm
We cannot divide by a variable unless we know it is not 0.

Question 16

What is a quadratic equation?

What form must it be in before we can factor it?

Answer 16

A quadratic equation is an equation in which the highest power of an unknown variable is 2.

It must be in ax^2 + bx + c = 0 (or general form) before we can factor it.

e.g., x^2 + 2x + 1 = 0
10x^2 = 100x
5c^2 + 10 = 10c - 20

Question 17

What are factors of a quadratic equation?

Which of the following is in factored form vs. general form:

x^2 + 4x + 4 = 0
(x + 2)(x + 2) = 0

Answer 17

A quadratic equation’s factors are expressions that are multiplied together to produce the original quadratic equation.

e.g., (x + 2)(x + 2) = 0 are the factors for x^2 + 4x + 4 = 0

Question 18

What is the difference between factors, roots, and solutions of a quadratic equation?

e.g., x^2 + 9x + 8 = 0

Answer 18

Using x^2 + 9x + 8 = 0 as the example:

Factors are what multiply to give us our original quadtratic.
Here, they are (x + 1) and (x + 8), giving (x + 1)(x + 8) = 0.

Roots and solutions are the same and they are the values that satisfy the equation. Here, they are x = -1 or x = -8.

We know these are the solutions because when multiplied, our product is zero, so one of these factors must equal 0.

Question 19

What does FOIL stand for?

Answer 19

When expanding quadratic factors into general form:
Using (x - 7)(x + 4) = 0 as an example:

First terms of each parenthesis (x * x = x^2)
Outer terms of each parenthesis (x * 4 = 4x)
Inside terms of each parenthesis (-7 * x = -7x)
Last terms of each parenthesis (-7 * 4 = -28)

x^2 + 4x - 7x -28 or x^2 - 3x -28

Question 20

What does (4x/x) equal?

Answer 20

4x/x is equal to 4.

Do not make the mistake of saying it equals 3x.

Question 21

What are the 3 common quadratic identities?

Answer 21

1. (x + y)^2 = (x+y)(x+y) = x^2 + y^2 + 2xy
2. (x - y)^2 = (x-y)(x-y) = x^2 + y^2 - 2xy
3. (x + y)(x - y) = x^2 - y^2

Question 22

What does x^2 + y^2 + 2xy equal?

Answer 22

(x + y)^2

Question 23

What does x^2 + y^2 - 2xy equal?

Answer 23

(x - y)^2

Question 24

What does x^2 - y^2 equal?

Answer 24

(x - y)(x + y)

Question 25

What is the Difference of Squares?

Answer 25

(x + y)(x - y) = x^2 - y^2

Question 26

What is a helpful way to recognize a difference of squares?

Answer 26

Look for one squared value minus another squared value.

e.g.,
x^2 - 9 = (x - 3)(x + 3)
x^2 - 1 = (x - 1)(x + 1)
4x^2 - 16 = (2x - 4)(2x + 4)
(5!)^2 - (4!)^2 =

Question 27

What is another way of writing this:

x^2 - 1

Answer 27

x^2 -1
= x^2 - 1^2
= (x - 1)(x + 1)

Question 28

What is another way to express the following:
-(x - y)

Answer 28

y - x

Question 29

(x - y)/(y - x) = ?

Answer 29

(x - y)/(y - x)
= (x - y)/-(x - y)
= -1

This is true no matter the x or y.

Question 30

(y - x)/(x - y) = ?

Answer 30

(y - x)/(x - y)
= -(x - y)/(x - y)
= -1

Question 31

What is a constant?

Answer 31

*A constant is a value that doesn’t change.

All quadratics have a constant term. In ax^2 + bx + c = 0, this is c.

Question 32

Review constant with quadratics.

Answer 32

From what I gather, you basically plug in the solution (using the given or the root) and solve for the constant.

Question 33

Which is greater:
(x + y)^2
or
(x^2 + y^2)

Answer 33

It depends upon the value of xy.

If xy is positive: (x + y)^2 is greater, because we’re adding 2xy.

If xy is negative: (x + y)^2 is less than, because we’re subtracting 2xy.

If the sign of xy is unknown: we cannot tell.

Question 34

Which is greater:
(x + y + z)^2
or
x^2 +y^2 + z^2

Answer 34

As x, y, and z are positive, (x + y + z)^2 is the greater sum because it translates to x^2 +y^2 + z^2 + 2xy + 2xz + 2yz.

Question 35

The six common equation traps?

Answer 35

1) Don’t assume that two equations are sufficient to determine the values of two variables, especially when one equation is just a multiple of the other.
(e.g., x + y = 2 and 10x + 10y = 20)

2) Don’t assume that the same number of unique equations is necessary to determine the values of those variables. We can at times solve for two variables when given only one equation.
(e.g., If x and y are positive integers and 3x + y = 5, then x = ?)

3) Don’t assume that we can’t determine a unique value for an expression (e.g., x+y, x-y, etc.) just because the expression has two unknown variables.
(e.g., if z + 2x - 8 = 2y + 4 + z, what is x - y?)

4) Don’t assume that if a quadratic equation solution has two unique roots, you may need more info. Similarly, don’t assume a one solution answer to a quadratic is not info.
(e.g., x^2 - 2x + 35 = 0 and x + y = 10; is A (x) or B (y) greater?)

5) Don’t assume an equation can’t have 3 or more solutions. Cubic (x^3) and quartic equations (x^4) can have 3 and four solutions, respectively.
(e.g., if x^4 = 36x^2, what are the solutions of x?)

6) Don’t assume that a variable or variable expression cannot be zero. You can only divide when you know for certain that the value of the variable or variable expression is not zero.
(e.g., 7x^2 = 35x)

Question 36

If we have two unique variables, but the two equations are the same, can we solve for the equation?

e.g.,
x + y = 8
and
(4/6)x + (4/6)y = 16/3

Answer 36

Sometimes.

Question 37

What is a cubic equation?

What is a quartic equation?

What are our options for solving them?

Answer 37

An equation where the highest power of an unknown variable is 3 is a cubic equation.

An equation where the highest power of an unknown variable is 4 is a quartic equation.

If both sides are not zero, we have to move all terms to one side before solving (similar to how we’d do with a quadratic).

We have two options:

Factoring out the Greatest Common Factor (GCF).
or
Factor by grouping.

Question 38

What are the ways we can solve cubic and quartic equations?

Answer 38

First, we must get one side of the equation equal to zero.

1. Factoring out the Greatest Common Factor (GCF):
- Useful when one side is equal to zero, and
- The other side has a common factor for every term.
(e.g., x^3 - 4x = 0)

2. Factoring by grouping:
- Useful when one side is equal to zero, and
- The other side has four terms, where two terms share a common factor and the other two share a common factor.
(e.g., x^3 - x^2 - 81x + 81 = 0)

Question 39

Can we divide by a variable?

Can we divide by an expression containing a variable?

If so, how do we know when we can?

Answer 39

Yes, we can, but only when we know that the variable does not equal zero.

e.g., Some of these can be divided by a variable (or variable expression), while others cannot.
x^2(x - 5) = 9 (x - 5)
x>0 and 17x^2y^3 - 23x^2z^5 = 0
7x^2 = 35x

Question 40

What is factoring by grouping?
When is it best used?
What conditions must we have?

Answer 40

Factoring by grouping is best used when we have a cubic or quartic equation (but can be used otherwise*).

It’s when we factor by groups and was introduced when we had four terms on one side, where the equation is set to zero, and two pairs of the variables had a common factor.

Question 41

What is factoring out the greatest common factor (GCF)?
When is it best used?
What conditions must we have?

Answer 41

It is best to factor out the GCF when we have a cubic or quartic equation (but can be used otherwise*).

It is when we pull out the GCF of the terms to solve for the variable.

We need a common factor for all of the terms and one side of the equation to be equal to zero.

Question 42

When solving for an unknown constant in a quadratic, how do we approach?

e.g., -2 is a solution x^2 - 6nx - 40 = 0…

what could be a product of n and x?

Answer 42

Generally, we can approach as follows:
1. Plug in our known solution for x
(e.g., (-2)^2 - (6)(-2)n - 40 = 0)

2. Solve for n and plug it back into the equation.
(e.g., n = 3 so x^2 + 18x - 40 = 0)

3. Break it out into factors
(e.g., (x - 20) (x + 2) = 0, so x = 20 or -2)

4. Use the information as it relates to the ask
(e.g., if n = 3 and x = 20 or -2, nx = 60 or -6)

Question 43

Note: don’t forget how to solve equations where we have two unknowns and we know they’re integers.

e.g., 5x + 8y = 55

Answer 43

5x + 8y = 55

8y = 55 - 5x
8y = 5 (11 - x)
y = [5 (11 - x)] / 8
Since we know they’re integers, and 5 isn’t divisible by 8, (11 - x) must be, so x must be 3.

Then, if x = 3, y = 5.