1. Equations with Radicals
Kathy Lyden's Equations with Radicals

2. Steps to Solving Equations with Radicals
Isolate the radical
Square both sides of the equation
(or raise to appropriate power)
Solve for the variable
Check all solutions. If the answer(s) does not work, the equation may not have a real number solution.
Note: You can also use the properties of
radicals to solve equations
Kathy Lyden's Equations with Radicals

3. Radical Equations EXAMPLE 1
use properties of radicals
isolate the variable
distributive property
rationalize the denominator
the conjugate of is
when the two binomials in the
denominator are multiplied out,
the two center terms cancel each other
Kathy Lyden's Equations with Radicals 3

4. Kathy Lyden's Equations with Radicals
EXAMPLE 1 page 2
expand the denominator
distribute the numerator
simplify the denominator
reduce by common factor
Check the solution in the original equation:
distributive property on the right
Kathy Lyden's Equations with Radicals 4

5. Radical Equations EXAMPLE 2
isolate the radical
square both sides
solve for the variable
Check the solution in the original equation:
The answer does not check in the
original equation, therefore there are
no real number solutions.
Kathy Lyden's Equations with Radicals

6. Radical Equations EXAMPLE 3
isolate the radical
square both sides
solve for the variable
Check the solution in the original equation:
Kathy Lyden's Equations with Radicals 6

7. Radical Equations EXAMPLE 4
isolate one of the radicals
square both sides
isolate the radical / /
Kathy Lyden's Equations with Radicals 7

8. Kathy Lyden's Equations with Radicals
EXAMPLE 4 page 2
solve for the variable
Check the solution in the original equation:
Kathy Lyden's Equations with Radicals 8

9. Equations with Radicals: EXERCISES
Solution
1. 2x = x = square both sides
2x = 100
x = (checks)
2. x + x = x + x = isolate the radical
x = - x square both sides
x = x isolate the variable
0 = x2 - x factor and solve
0 = x(x - 1)
x = 0 and x = (only 0 checks)
Kathy Lyden's Equations with Radicals

10. Kathy Lyden's Equations with Radicals
Equations with Radicals: EXERCISES 2
Solution
y - 7 = y - 7 = cube both sides
y - 7 = 27
y = (checks)
4. x - x - 1 = x x - 1 = isolate the radical
- x - 1 = - x square both sides
x - 1 = x2 - 6x isolate the variable
0 = x2 - 7x factor and solve
0 = (x - 5)(x -2)
x = 5 and x = (only 5 checks)
Kathy Lyden's Equations with Radicals

11. Kathy Lyden's Equations with Radicals
Equations with Radicals: EXERCISES 3
Solution
x x - 3 = x - 7 = x isolate one radical
x - 7 = x x square both sides
- 8 = x isolate the radical
2 = x square both sides
4 = x solve for the variable
7 = x (checks)
6. r = , for d r = cube both sides
r3 = isolate the variable d
r3 = 3w
d = (checks) / / 3 3w 3 3w
4pd
4pd 3w
4pd
4pd
3 w
4pr3
Kathy Lyden's Equations with Radicals