P.5 Solving Equations
Objectives: Solve linear, quadratic, cubic and higher degree polynomial equations, radical and absolute
value equations.
Linear Equations
Define as many of the following as you can
1.
2.
3.
4.
5.
6.
7.
8.









Equation
Solve
Solutions
Identity
Conditional Equation
Linear Equation
Equivalent Equations
Extraneous Solutions
Equation: a statement that two algebraic expressions are =
o 3x – 5 = 7
Solve: find all the values of x (or the variable) that make the equation true
Solutions: The found values for which the statement is true. Can always be used to check.
Identity: The same thing on both sides of the equation. In this case you can plug in absolutely
any number and it would be true. Example: 3x + 4x = 7x
Conditional Equation: has either no solution or is limited to only having certain ones. Basically,
the ones you’re used to seeing most often. Examples 3x – 5 = 7, x^2 – 9 = 0, 3x + 5 = 3x + 2
Linear Equation: only has one variable and can be written in the standard form: ax + b = 0, or mx
+ b is the same thing. Don’t get tripped up on the one variable thing when we say y = mx + b,
now it’s a function, as long as there is no exponent other than 1 on x, it’s linear.
Equivalent Equations: Two equations that have the same solutions
Extraneous Solutions: solutions that appear true, but do not satisfy the original equation. Always
check your answers to avoid giving these as your answers.
Examples:
o 3x – 6 = 0
o 8x – 5 = 3x + 20
o 7x – 7 = 7(x – 1)
o
𝑥
3
o
3
7
3
+ 𝑥+1 = 𝑥
𝑥(𝑥+1)
1
3
6𝑥
=
− 2
𝑥−2
𝑥+2
𝑥 −4
o

+
3𝑥
4
=2
o Hw: 16 – 40 evens
Quadratic Equations aka second-degree polynomial: ax^2 + bx + c = 0
o Methods to solve
 Factor, then set factors = 0



 𝑥2 − 𝑥 − 6 = 0
 2𝑥 2 + 9𝑥 + 7 = 3
 6𝑥 2 − 3𝑥 = 0
 Take the square root, whatever is being squared must be isolated
 4𝑥 2 = 12
 (𝑥 + 3)2 = 16
 (𝑥 + 5)2 + 10 = 35
 Completing the square (book calls it ‘extracting square roots’), a must be one
 𝑥 2 + 6𝑥 = 5
 𝑥 2 − 2𝑥 − 3 = 0
 Quadratic Formula
 2𝑥 2 + 3𝑥 − 1 = 0
*****Make sure to make the point that although not all can be factored or done by isolating the
square root, all can be solved by completing the square and by using the quadratic formula.
 Quadratic Equation homework: 56 – 66 evens, 74 – 80 evens, 102 – 110 evens,
128, 130
When solving higher degree polynomial equations, simply factor then solve
o 3𝑥 4 = 48𝑥 2
o 𝑥 3 − 3𝑥 2 − 3𝑥 + 9 = 0
o Hw: 142 – 146
Radical Equations- involve either radical (square root) signs, or fractional exponents
o Examples
3

4𝑥 2 − 8 = 0 to get rid of the fractional exponent, first get that variable by
itself, then raise it to it’s reciprocal.


(𝑥 + 2)3 = 9
√2𝑥 + 7 − 𝑥 = 2, to get rid of the radical in an equation, isolate it then square
it (show the kids that this is the same as raising it to it’s reciprocal.
2
 √2𝑥 − 5 – √𝑥 − 3 = 1
 Hw: 153 – 156, 163 – 167, 171 - 177
 Absolute Value Equations |x|= 2, what’s the value of x? 2 and -2, because there is always two
possibilities, we always set up two equations. Always make sure the abs. value bars are alone
first. Always check for extraneous solutions.
o |2x + 3| + 5 = 10
o |x – 2|=3
o |𝑥 2 − 3𝑥| = −4𝑥 + 6
o Hw: 179, 181, 183, 185 – 187
Homework: 16 – 40 evens, 56 – 66 evens, 74 – 80 evens, 102 – 110 evens, 128, 130, 142 – 146, 153 –
156, 163 – 167, 171 – 183 odds, 185- 187