Math1000 Section1.5

1. Math 1000
Stuart Jones
Section 1.5
Equations

2. Math 1000
Stuart Jones
In this section, we will look at a wide variety of different types
of equations. Each of these will wiggle their way into your
mathematical future. The first we will look at is a linear
equation. Linear equations can have one solution, no solution,
or ALL REAL NUMBERS as solutions. Let’s look at a few.

3. Math 1000
Stuart Jones
Solve
3x + 4 = 16

4. Math 1000
Stuart Jones
Solve
−2(3x + 4) = 2(x − 4) − 8x

5. Math 1000
Stuart Jones
Solve
−2x + 7 = 2(−x − 5)

6. Math 1000
Stuart Jones
The second type of equations we will look at are sometimes
called literal equations. In this case, we are asked to solve a
given equation for a certain variable.

7. Math 1000
Stuart Jones
Solve V = πr2h for r

8. Math 1000
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Solve W = 4vr2s
3p for p

9. Math 1000
Stuart Jones
The nest group of equations, quadratic equations, is a rather
large group. We will study 3 different methods of solving them.
The first, and sometimes easiest, is by factoring. To use this,
we also need an added tidbit:
Zero Product Property
ab = 0 ⇐⇒ a = 0 or b = 0

10. Math 1000
Stuart Jones
Solve by factoring:
x2
+ 2x − 63 = 0

11. Math 1000
Stuart Jones
Solve by factoring:
x2
− 81 = 0

12. Math 1000
Stuart Jones
Completing the Square
Start with a quadratic equation ax2 + bx + c = 0.
1 If the leading coefficient is not 1, divide both sides by a,
the leading coefficient, to make it 1.
2 Move the constant term to the right side.
3 Take the b term (whatever it is after dividing by a), and
divide it by 2. Remember this number.
4 Take the number from the previous step, and square it.
5 Add the number from the previous step to both sides of
the equation.
6 Rewrite the left side of the equation as (x + @)2, where @
is the number that you divided by 2 above.
7 Simplify the right side, then square root both sides and
move the constant to get x by itself.

13. Math 1000
Stuart Jones
Solve by Completing the Square:
x2
+ 6x − 55 = 0

14. Math 1000
Stuart Jones
Solve by Completing the Square:
x2
+ 6x − 55 = 0
For time’s sake, this is the only example we’ll do in the notes,
but please do some of the homework problems on this! See me
for help on this.

15. Math 1000
Stuart Jones
Finally, with quadratic equations, we have the Quadratic
Formula:
Quadratic Formula
If we have ax2 + bx + c = 0, then:
x =
−b ±
√
b2 − 4ac
2a

16. Math 1000
Stuart Jones
Solve by the Quadratic Formula:
x2
− 6x + 2 = 0

17. Math 1000
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The next type of equation we will look at is a rational
equation. The general strategy here is going to be to multiply
to get rid of our denominators first.

18. Math 1000
Stuart Jones
Solve:
x
x − 1
+
x
x + 1
= 0

19. Math 1000
Stuart Jones
The last type of equation we will look at this section is radical
equations. Here, the goal is to get a radical by itself, then get
rid of it, and if needed, repeat. MAKE SURE to check your
answer here, because it’s possible you could get fake answers
(called extraneous) doing this.

20. Math 1000
Stuart Jones
Solve:
2x +
√
x + 1 = 1

21. Math 1000
Stuart Jones
The Bottom Line
This is a big section to highlight two points you’ll see
throughout this class - MEMORIZE the
formulas/procedures (they will not be given to you on the
test), and practice! Do the homework problems. If you do
not, you will likely struggle with this.