Section 9 – 7:
Solving Radical Equations
Selected Worked Homework Problems
How to Solve Radical Equations with Square Roots
1. Get the term with the square root alone one one side of the equation.
2. If the isolated square root is equal to a negative number then STOP. There are No Real Numbers
that will work. We write NRN as the answer. We have solved the equation and do not go on.
3. Square both sides of the equation.
4. Solve for x.
5. Check your answer(s). If no answers work then there are No Real Number that are solutions.
Solve each Radical Equation for x. Use NRN if needed.
1.
x−3=5!
2.
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
The square root is alone
on one side of the equation
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
(
x−3
)2 = 52
2x − 1 = 7
(
)2 = 72
2x − 1
x − 3 = 25 solve for x
2x − 1 = 49 solve for x
2x = 50
x = 25
x = 28
Check:
!
2(25) − 1 = 7
Check:
28 − 5 = 5
49 = 7
7= 7
25 = 5
5=5
!
Math 100 !
Section 9 – 7 HW WKD!
© 2016 Eitel
3.
2x + 15 = −5 !
The square root is alone
on one side of the equation
but the number it is equal
to is NEGITIVE.
4.
5x + 1 − 7 = −4 !
Get the square root alone
on one side of the equation
by adding 7 to both sides
5x + 1 − 7 + 7 = −4 + 7
STOP: An isolated square root
cannot equal a negitive number
so the answer is NRN
Had you continued you would
have found that
x =5
!
but that answer would not have
been true if you checked it by
putting it back into the equation.
2(5) + 15 ≠ −5
10 + 15 ≠ −5
25 ≠ −5
5 ≠ −5
5x + 1 = 3
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
(
)2 = 32
5x + 1
5x + 1 = 9 solve for x
5x = 8
8
x=
5
Check:
⎛ 8⎞
5
+ 1 − 7 = −4
⎝ 5⎠
8 + 1 − 7 = −4
9 − 7 = −4
3 − 7 = −4
Math 100 !
Section 9 – 7 HW WKD!
© 2016 Eitel
5.
7x + 8 + 4 = 10 !
4 x + 12 − 6 = −6
6.
Get the square root alone
on one side of the equation
by subtracting 4 from both sides
Get the square root alone
on one side of the equation
by adding 6 to both sides
7x + 8 + 4 = 10
− 4 = −4
4x + 12 − 6 + 6 = −6 + 6
7x + 8 = 6
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
(
7x + 8
)
2
= 62
7x + 8 = 36 solve for x
7x=26
x=4
4x + 12 = 0
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
!
(
4x + 12
)
2
= 02
4x + 12 = 0 solve for x
4x= − 12
x= − 3
Check:
4(−3) + 12 − 6 = −6
−12 + 12 − 6 = −6
0 − 6 = −6
0 − 6 = −6
−6 = −6
Math 100 !
Section 9 – 7 HW WKD!
© 2016 Eitel
18.
2x −1 + 4 = 3!
Get the square root alone
on one side of the equation
by subtracting 4 from both sides
−3x + 1 − 5 = −1
22.
Get the square root alone
on one side of the equation
by adding 5 to both sides
−3x + 1 − 5 + 5 = −1 + 5
2x − 1 + 4 − 4 = 3 − 4
2x − 1 = −1
The square root is alone
on one side of the equation
but the number it is equal
to is NEGITIVE.
STOP: An isolated square root !
cannot equal a negitive number
so the answer is NRN
−3x + 1 = 4
The square root is alone
on one side of the equation
and the number it is equal
to is NOT NEGITIVE
so square both sides
(
)2 = 4 2
−3x + 1
−3x + 1 = 16 solve for x
−3x = 15
x = −5
Check:
−3x + 1 − 5 = −1
−3(−5) + 1 − 5 = −1
15 + 1 − 5 = −1
16 − 5 = −1
4 − 5 = −1
−1 = −1
Math 100 !
Section 9 – 7 HW WKD!
© 2016 Eitel
3x + 20 = x + 28 !
31.
The square roots are alone
on eash side of the equation
so square both sides.
The square roots are alone
on eash side of the equation
so square both sides
(
3x + 20
) =(
2
x + 28
x + 1 = 3x − 5 !
32.
)
(
2
3 + 1 = 3(3) − 5
3(4) + 20 = 4 + 28
4 = 9−5
32 = 32
4= 4
−x + 3 = −2x + 1!
The square roots are alone
on eash side of the equation
so square both sides.
(
−x + 3
)2
Check:
Check:
) (
2
3x − 5
x + 1 = 3x − 5 solve for x
6 = 2x
3=x
3x + 20 = x + 28 solve for x !
2x = 8
x=4
34.
)2 = (
x +1
=
−2x + 1
)
2
−x + 3 = −2x + 1 solve for x
x =−2
!
Check:
−(−2) + 3 = −2(−2) + 1
2 + 3 = 4 +1
5= 5
Math 100 !
2x 2 − 3x − 4 = x
36.
Square both sides.
(
2x 2 − 3x − 4
) =x
2
2
2x 2 − 3x − 4 = x 2 solve for x
x 2 − 3x − 4 = 0
(x − 4)(x + 1) = 0
x = 4 x = −1
When you plug 4 in it works
but when you plug x = − 1
in it does not work
So the only answer is x = 4
Section 9 – 7 HW WKD!
© 2016 Eitel