1. Aim #1.6: How do we solve other types of equations?

2. In this section we will learn how to solve:
Other types of Polynomial Equations
Radical Equations
Equations w/ Rational Exponents
Equations that are Quadratic Form
Absolute Value Equations

4. Example 1: Solve a Polynomial Equation by Factoring
3𝑥 4 = 27𝑥 2

5. Example 2: Solve a Polynomial Equation by Factoring
Solve by Factoring:
𝑥 3 + 𝑥 2 =4𝑥+4

6. Check for Understanding:
Solve by factoring:
𝟒𝒙 𝟒 = 𝟏𝟐𝒙 𝟐
𝟐𝒙 𝟑 +𝟑 𝒙 𝟐 =8x + 12

7. Radical Equations:
A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve the equation by squaring both sides.

8. This new equation has two solutions, -4 and 4
This new equation has two solutions, -4 and 4. By contrast, only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, check proposed solutions in the original equation.
Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation are called extraneous solutions or extraneous roots.

9. How do we solve Radical Equations?

10. Example 3: Solving a Radical Equation
Solve:

11. Check for Understanding:
Solve: 2𝑥−1 +2=𝑥

12. Solving an Equation That Has Two Radicals
Isolate a radical on one side.
Square both sides.
Repeat Step 1: Isolate the remaining radical on one side.
Repeat Step2: Square both sides.
Solve the resulting equation
Check the proposed solutions in the original equations.

13. Example 4: Solving an Equation with Two Radicals
Solve: 3𝑥+1 − 𝑥+4 =1

14. Check for Understanding:
Solve:

15. Equations w/ Rational Exponents

16. Example 5: Solving Equations w/ Rational Exponents
Solve:
3 𝑥 −6=0
𝑥 − 3 4 =− 1 2

17. Check for Understanding:
Solve:

18. Equations that are in Quadratic Form:

19. Equation 6: Solving an Equation Quadratic in Form:

20. Check for Understanding:
𝑆𝑜𝑙𝑣𝑒: 𝑥 4 −8 𝑥 2 −9=0

21. Example 7: Solving an Equation Quadratic in Form:
Solve: 𝑥 𝑥 =0

22. Check for Understanding:
Solve: 𝑥 𝑥 −4=0

23. Example 8: Equations Involving Absolute Value

24. Check for Understanding:
Solve: 2𝑥−1 =5

25. Example 9: Equations Involving Absolute Value
Solve: 5 1−4𝑥 −15=0

26. Check for Understanding:
Solve: 4 1−2𝑥 −20=0

27. Review:
Solve and Check your solution:

28. Review:
Solve:

29. Summary: Answer in a complete sentence.
In solving
Why is it a good idea to isolate a radical term?
What is an extraneous solution to a radical equation?