The Square Root Property

The Square Root Property
Recall:
π’™πŸ = πŸ—
Since this is a quadratic equation, we must set it equal to zero,
factor, set each factor equal to zero, and solve.
π’™πŸ = πŸ—
π’™πŸ βˆ’ πŸ— = 𝟎
(𝒙 + πŸ‘)(𝒙 βˆ’ πŸ‘) = 𝟎
𝒙+πŸ‘=𝟎
𝒙 = βˆ’πŸ‘
π’™βˆ’πŸ‘=𝟎
𝒙=πŸ‘
Solving the equation mentally, we should ask ourselves, β€œwhat
number when squared gives us πŸ—?”
πŸ‘ squared is πŸ— and (βˆ’πŸ‘) squared is πŸ—.
We know we should get 𝟐 answers for a quadratic. When we want to
solve this equation we can square root both sides, but we must keep in
mind that there are two possible results: one positive and one negative.
π’™πŸ = πŸ—
βˆšπ’™πŸ = ±βˆšπŸ—
𝒙 = ±πŸ‘
Note: Recall that the symbol ± means β€œplus or minus”. When
taking the square root of both sides of an equation, we add ± to
indicate both the positive AND negative solution.
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Example 1:
Solve using the square root property.
a.) π’™πŸ = πŸπŸ”
b.) π’™πŸ = 𝟏𝟐
c.) π’™πŸ + 𝟏 = πŸπŸ”
Here we want to isolate the π’™πŸ so we can use the square root
property.
π‘₯ 2 + 1 = 26
Subtract 𝟏
π’™πŸ = πŸπŸ“
Square root prop.
βˆšπ’™πŸ = ±βˆšπŸπŸ“
Simplify
𝒙 = ±πŸ“
Example 2:
Solve using the square root property.
a.) (𝒙 + 𝟏)𝟐 = πŸ–
(π‘₯ + 1)2 = 8
√(𝒙 + 𝟏)𝟐 = ±βˆšπŸ–
Square root prop.
𝒙 + 𝟏 = ±βˆšπŸ–
Simplify
𝒙 + 𝟏 = ±πŸβˆšπŸ
Simplify
𝒙 = βˆ’πŸ ± 𝟐√𝟐
Subtract 𝟏
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b.) πŸ‘π’™πŸ βˆ’ 𝟏 = πŸπŸ’
πŸ‘π’™πŸ βˆ’ 𝟏 = πŸπŸ’
Add 𝟏
πŸ‘π’™πŸ = πŸπŸ“
Divide πŸ‘
π’™πŸ = πŸ“
Square root prop.
βˆšπ’™πŸ = ±βˆšπŸ“
Simplify
𝒙 = ±βˆšπŸ“
c.) πŸ•π’™πŸ βˆ’ 𝟏𝟎 = πŸ’πŸ”
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The Square Root Property
Practice Problems
Solve each equation using the square root property.
1. π’™πŸ = 𝟏𝟎𝟎
2. π’™πŸ + πŸ“ = πŸ‘πŸŽ
3. πŸπ’™πŸ βˆ’ πŸ‘ = 𝟐𝟏
4. (𝒙 + πŸ‘)𝟐 = πŸ—
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