1. MA 1128: Lecture 17 – 6/17/15 Adding Radicals Radical Equations

2. Adding/Subtracting Radicals Be very careful when you see addition or subtraction mixed with radicals. For example, Next Slide and Are not the same!! You can see this by simplifying each of them. These are clearly not equal.

3. Special Situations We can simplify in a few special situations (and otherwise, we’ll leave them as they are). Both of the situations mentioned here involve factoring out. This first example is like combining like terms and uses the distributive property. The radical parts have to be exactly the same. Next Slide This next example factors inside the radical, and depends on exponents and radicals distributing over multiplication or division. By factoring inside, we get multiplication under the radical, and radicals split with multiplication.

4. Practice Problems Simplify if you can. Next Slide Answers on next slide.

5. Answers Simplify if you can. Next Slide There isn’t much we can do with the expressions in problems 3 and 4

6. Radical Equations We have a little more freedom when we’re working with equations. With an equation, we can do the same thing to both sides. For example, we can square both sides of this next equation. Next Slide Since taking a square root and squaring are inverse operations, they undo each other.

7. Be sure you check your answers If you check x = 8 in this last equation, you’ll see that it works. You do have to be careful with squaring both sides of an equation, however. Squaring can make unequal things equal. For example, 3  -3, but (3) 2 = (-3) 2. Note that the right side of this next equation is negative. Next Slide Squaring both sides of an equation will sometimes introduce wrong answers. You should always check your answers, and throw out the bad ones.

8. The radical should be by itself In order to get a simpler equation, you need to be sure that the radical is by itself on one side. For example, note what happens if you don’t do this. Next Slide There’s nothing really wrong with this, But we still have a radical, so we’re no better off. In fact, things got worse.

9. Example (cont.) WE SHOULD HAVE DONE THE FOLLOWING. Next Slide Check:This solution is fine.

10. Practice Problems Next Slide Answers: 1)x = 23 2)x = 7 3)No solutions. If you square both sides, you’ll end up with x = 7, but if you plug this back in, this solution doesn’t work. You can see this from the beginning, since the square root symbol is defined to indicate the positive square root, and the positive square root can’t be negative 3.

11. More Examples Be sure to square (or cube etc.) each side as a whole. You should square the entire left side and square the entire right side, not the individual terms. Look at the right side in the second line. Next Slide

12. More Examples You might need to cube (or something else). Next Slide

13. Practice Problems Solve the following equations. Next Slide Answers: 1)After squaring, you get a quadratic equation. Move everything to the right side, and you get 0 = x 2 – 7x – 8. This factors to 0 = (x + 1)(x – 8), so the solutions are x = -1,8. 2)x = 5

14. Two Radicals in an Equation Radical equations can be really hard, or even impossible, to do. The worst that we’ll consider are like the equations in the last two quiz problems, or equations with a single radical on both sides of the equation. The idea is basically the same, and they’re maybe even a bit easier. Just be sure that each radical is by itself on one side of the equation. Next Slide

15. Practice Problem End Solve. Answer 1) x =  1.