MA 1128: Lecture 17 – 4/12/11

Adding Radicals

Radical Equations

Adding/Subtracting Radicals

Be very careful when you see addition or subtraction mixed with radicals.

For example,

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169 and 169

Are not the same!!

You can see this by simplifying each of them.

743169

525169

These are clearly not equal.

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.

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333 222527

This next example factors inside the radical, and depends on exponents and radicals distributing over multiplication or division.

xxxx 322324)32(4128

By factoring inside, we get multiplication under the radical, and radicals split with multiplication.

Practice Problems

Simplify if you can.

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25.4

7.3

.2

323537.123

x

xx

Answers on next slide.

Answers

Simplify if you can.

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25.4

7.3

1)1(.2

310323537.1223

x

xxxxxx

There isn’t much we can do with the expressions in problems 3 and 4

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.

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2231

31

x

x

Since taking a square root and squaring are inverse operations,they undo each other.

8

91

x

x

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.

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3391)8(

8

91

)3(1

312

x

x

x

x

Squaring both sides of an equationwill sometimes introduce wrong answers.

You should always check your answers,and throw out the bad ones.

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.

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3625310)3(

362535533

365353

653

653

2

22

xx

xxx

xx

x

x

There’s nothing really wrong with this,

But we still have a radical, so we’re no better off. In fact, things got worse.

Example (cont.)

WE SHOULD HAVE DONE THE FOLLOWING.

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4

13

13

13

653

22

x

x

x

x

x

Check: 6515153)4( This solution is fine.

Practice Problems

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32.3

423.2

52.1

x

x

x

Answers:1) x = 232) x = 73) 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.

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.

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x

x

x

xxx

xx

xx

1

22

123

123

13

13

22

22

2

2

More Examples

You might need to cube (or something else).

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5

306

2736

336

336

333

3

x

x

x

x

x

Practice Problems

Solve the following equations.

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213.2

2123.1

4

x

xx

Answers:1) After squaring, you get a quadratic equation. Move everything to the right side, and you get 0 = x2 – 7x – 8. This factors to 0 = (x + 1)(x – 8), so the solutions are x = -1,8.2) x = 5

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.

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6

2748

2748

274822

x

xx

xx

xx

x

xx

xx

xx

xx

8

28

28

28

028

44

44

44

44

Practice Problem

End

33 83105.10 xx

Solve.

Answer1) x = 1.