Algebra 1
Chapter 10: Square Root Equations
Name: ______________________________
1. As we learned when working through the Pythagorean Theorem, squares and square roots
are inverse operations, they “undo” each other. When solving an equation with a square,
we take the square root on both sides. When solving an equation with a square root, we
square each side. Solve the following radical equations by squaring both sides.
a.
𝒙=𝟕
b.
𝒚=𝟓
c.
𝒎 = 𝟐𝟓
2. Sometimes there is more under the radical than just the variable. But because the square
and square root “undo” each other, the value under the radical doesn’t change. For
instance, if I square 𝟑𝒃 + 𝟒 I will just get 3b + 4. Solve the following radical equations
while showing work.
a.
𝟒𝒂 = 𝟖
b.
𝟓𝒏 = 𝟏𝟎
c.
d.
𝟓𝒙 − 𝟏 = 𝟕
e.
𝟐𝒃 + 𝟖 = 𝟔
f.
𝟐𝒌 = 𝟓
1 = −𝟐𝒗 − 𝟑
3. The equations in problems #1 and 2 all had isolated radicals (square roots that were by
themselves on one side of the equation). If the radical isn’t isolated, you need to do that
first, then square both sides. Solve these one while showing work. Be sure to get the radical
by itself first, then square both sides.
a.
𝒂 + 𝟑 = 𝟐𝟐
b.
𝒏 − 𝟔 = 𝟏𝟎
c.
𝒌 + 𝟑𝟏 = 𝟓
d.
𝟓𝒂 + 𝟕 = 𝟏𝟕
e.
𝟐𝒓 + 𝟒 = 𝟏𝟔
f.
3 − 𝒚 = −𝟐
4. Sometimes there are radicals on both sides of the equations. Simply start by squaring both
sides, then solving the resulting equation.
a.
𝟑𝒙 + 𝟏 = 𝟓𝒙 − 𝟖
b.
𝟐𝒚 =
𝟗−𝒚
d.
𝒔 + 𝟏𝟎 = 𝟔 − 𝒔
e.
𝒏 + 𝟓 = 𝟓𝒏 − 𝟏𝟏
c.
𝟕𝒗 − 𝟒 = 𝟓𝒗 + 𝟏𝟎
f.
𝟑𝒎 + 𝟏 = 𝟕𝒎 − 𝟗
New Topic that also involves radicals:
5. Recall collecting like terms. Simplify the following
a.
𝟓𝒙 + 𝟑𝒙
b.
𝟏𝟎𝒚 − 𝟒𝒚 c.
9𝒙 + 𝟑𝒚 − 𝟓𝒙 + 𝟐𝒚
6.
If I tell you that 𝒙 = 𝟐 and 𝒚 = 𝟓, then the above expressions could be written as the
expressions shown below. The mathematics of simplifying these is the same as for #5, add the
coefficients. Simplify these problems then check them on a calculator.
a.
𝟓 𝟐+𝟑 𝟐
𝟏𝟎 𝟓 − 𝟒 𝟓 b.
c.
9 𝟐+𝟑 𝟓−𝟓 𝟐+𝟐 𝟓
7. Simplify the following radical expressions by collecting like terms.
a.
𝟓+𝟑 𝟓
b. 𝟏𝟐 𝟕 − 𝟗 𝟕
c.
𝟕 𝟑+ 𝟑
d.
𝟏𝟏 + 𝟒 𝟓 − 𝟐 𝟏𝟏
e.
𝟔 − 𝟗 𝟕 + 𝟑 𝟔 − 𝟒 𝟕 f.
𝟏𝟎𝟓 𝟑𝟏 + 𝟗𝟕 𝟑𝟏