BPM MATH0 – Week 4
Radical Expressions and Equations
Week 4 – Radical Expressions and Equations
Weekly Goals
ˆ Simplify expressions involving square roots.
ˆ Rationalize denominators with one or more radicals.
ˆ Solve equations that include radicals.
Solved Examples – With Detailed Steps
Example 1: Simplify:
1
√
x − 2
+
1
√
x + 2
Steps:
ˆ Use a common denominator:
√
x + 2 +
√
x − 2
x − 4
=
2
√
x
x − 4
2
√
x
x − 4
Example 2: Rationalize the denominator:
3
√
5 −
√
2
Steps:
ˆ Multiply numerator and denominator by the conjugate:
3
√
5 −
√
2
·
√
5 +
√
2
√
5 +
√
2
=
3(
√
5 +
√
2)
5 − 2
3(
√
5 +
√
2)
3
=
√
5 +
√
2
Example 3: Solve the equation:
√
x + 1 = x − 1
Steps:
ˆ Square both sides: x + 1 = x2
− 2x + 1
ˆ Rearranged: 0 = x2
− 3x
ˆ Factor: x(x − 3) = 0 ⇒ x = 0 or x = 3
BPM MATH0 – Week 4
Radical Expressions and Equations
ˆ Check both: only x = 3 satisfies original equation
x = 3
Example 4: Solve the equation:
√
x + 2 +
√
x − 1 = 5
Steps:
ˆ Let’s isolate one root: √
x + 2 = 5 −
√
x − 1
ˆ Square both sides:
x + 2 = 25 − 10
√
x − 1 + (x − 1) ⇒ x + 2 = x + 24 − 10
√
x − 1
ˆ Subtract x and 24:
−22 = −10
√
x − 1 ⇒
√
x − 1 =
11
5
ˆ Square again:
x − 1 =
11
5
2
=
121
25
⇒ x =
146
25
ˆ Final check: plug into original equation to verify.
x =
146
25
Practice Problems for Seminar
Simplify
1.
√
50 +
√
8
2. 2√
3
+ 1√
12
3. 3
2+
√
5
4.
√
x−2
x−4
Solve
5.
√
x + 4 = 3
6.
√
2x − 1 = x
7. 1 +
√
x + 2 = 2x
8.
√
x − 3 +
√
x + 1 = 4
9. Determine the domain of the expression f(x) =
√
x − 2 + 1√
5−x