Download Radical Equations

Radical Equations
A Radical equation is
an equation whose
unknown quantity
appears in the
radicand.
Consider the two sets of
equations below.
1. a. 3𝑥 + 2 = 14
b. 5 𝑥 + 2 = 30
2. a. 3 𝑥 + 2 = 14
b. 5 𝑥 + 2 = 30
 In solving radical equations, you
assume that if two numbers are
equal, then their squares, cubes, or
nth power are also equal.
𝑛
𝑛
 In symbol: If 𝑥 = 𝑦, 𝑡ℎ𝑒𝑛 𝑥 = 𝑦
for every positive integer n.
Steps in Solving Radical Equations
 Write the equation such that the radical
containing the unknown is on one side of the
equation.
 Combine similar terms.
 Raise both members of the equation to a
power whose exponent is the same as the
index of the radical. If the equation is free of
radicals, then complete the solution.
 Always check if the values obtained are
solutions of original equation.
1. Solve 𝑥 = 7 .
Solution:
𝑥 2 = (7)2 Square both sides
𝑥 = 49
Checking:
𝑥=7
49 = 7
7=7
3
2. Solve 𝑥 − 4 = 2 .
Solution:
3
𝑥−4
3
= 2
𝑥−4=8
=𝑥 =8+4
𝑥 = 12
Checking:
3
𝑥−4=2
3
12 − 4 = 2
3
8=2
2=2
3
Cube both sides, since the
index of the radical
expression is 3.
By APE
3. Solve 4 + 𝑥 − 2 = 𝑥 .
Solution:
𝑥−2=𝑥−4
Isolate the radical expression.
2
𝑥−2 = 𝑥−4 2
𝑥 − 2 = 𝑥 2 − 8𝑥 + 16
𝑥 2 − 9𝑥 + 18 = 0
𝑥−6 𝑥−3 =0
𝑥 − 6 = 0 𝑜𝑟 𝑥 − 3 = 0
Square both sides.
Combine similar terms.
Solve quadratic equation
by factoring.
by Zero Product Property
Checking:
If 𝑥 = 3
4+ 𝑥−2=𝑥
4+ 3−2=3
4+ 1=3
4+1=3
5≠3
If 𝑥 = 6
4+ 𝑥−2=𝑥
4+ 6−2=6
4+ 4=6
4+2=6
6=6
4. Solve for x: 3𝑥 − 5 = 𝑥 + 19 .
Solution:
2
2
3𝑥 − 5 = 𝑥 + 19
Square both sides.
3𝑥 − 5 = 𝑥 + 19
3𝑥 − 𝑥 = 19 + 5
Combine similar terms.
2𝑥 = 24
𝑥 = 12
by MPE
Checking:
3𝑥 − 5 = 𝑥 + 19
3 12 − 5 = 12 + 19
36 − 5 = 12 + 19
31 = 31
6. A man walks 4 meters to the west, and then
walks 9 meters northward. How far is the man from the
starting point?
Solution: Sketch or picture out the problem
9 m northward
The distance of the man from the
starting point is the hypotenuse of the
right triangle.
Starting
point
4 m west
c2 = a2+b2
c = a2+b2
=42+92
= 16+81
c = 97m,
The distance of
the man from the starting point or
approximately 9.8 m from the starting
point.
Solve for x.
1. 𝑥 = 15
2. 𝑥 + 10 = 12
3. 𝑥 + 7 = 5
4. 3𝑥 + 8 = 2𝑥 + 12
5. 𝑥 + 54 = 7 𝑥