Week 17: Radicals, domain, and elimination

Field Book

64 to 67

Due

Test 15 due

Method

Paper first, every step

The last full week before the midterm, and it earns its place. Simplifying radicals is the skill that makes every later answer look finished instead of half-done. Elimination is the second way to solve a system, and it is usually the faster one. Locke would point out that having two methods is not redundancy, it is having a spare.

What you have to be able to do

State the domain of an expression
Simplify a radical and add like radicals
Solve a system by elimination

The procedures

Simplifying a radical

Find the largest perfect square that divides the number under the sign.
Split it: the root of that product becomes two roots multiplied.
Take the root of the perfect square out front and leave the rest inside.

Adding radicals

Only like radicals combine. Like means the same number under the sign.
Simplify each radical first. Two that looked different often turn out to match.
Then add the numbers in front, exactly the way like terms work.

Elimination

Line the two equations up so matching variables sit above each other.
Multiply one or both equations so one variable has opposite coefficients.
Add the equations. That variable disappears. Solve for what is left, then back-substitute.

Worked

WORKED

Simplify the square root of 50.
25 is the largest perfect square dividing 50, and 50 = 25 × 2.
The root of 25 is 5, and the 2 stays inside.
ANSWER: 5 times the square root of 2

WORKED

Solve by elimination: x + y = 10 and x − y = 2.
The y terms are already opposites. Add the two equations: 2x = 12.
x = 6. Put it back into the first equation: 6 + y = 10.
ANSWER: x = 6, y = 4

Terms

Term	Meaning
domain	every value the variable is allowed to be
like radicals	radicals with the same number under the sign
elimination	adding two equations so one variable cancels out

Check yourself

1. Simplify the square root of 18

2. Three roots of 2 plus five roots of 2

3. Solve: x + y = 9 and x − y = 1

reveal answers

3 roots of 2 · 8 roots of 2 · x = 5, y = 4

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.