Algebra 1 Practice - Solving a Radical Equation (Example 2)

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*Algebra 1 practice on solving radical equations* focuses on solving equations that involve square roots or other radicals. These problems emphasize isolating the radical and eliminating it by raising both sides of the equation to the appropriate power.

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*Steps to Solve a Radical Equation*

1. *Isolate the radical expression.*
Move terms so that the radical is alone on one side of the equation.

2. *Eliminate the radical.*
Raise both sides of the equation to the power that matches the radical (e.g., square both sides for a square root).

3. *Solve the resulting equation.*
After removing the radical, solve the equation using algebraic techniques.

4. *Check your solutions.*
Substitute your solutions back into the original equation to ensure they do not create extraneous solutions.

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*Example Problems*

#### *Example 1: Solving with a Single Radical*
Solve:
\[
\sqrt{x + 3} = 5
\]

*Solution:*
Isolate the square root (already done).
Square both sides to eliminate the radical:
\[
(\sqrt{x + 3})^2 = 5^2
\]
\[
x + 3 = 25
\]
Solve for \(x\):
\[
x = 25 - 3 = 22
\]
Check:
\[
\sqrt{22 + 3} = \sqrt{25} = 5
\]
Valid solution.

Final answer:
\[
x = 22
\]

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#### *Example 2: Solving with Two Radicals*
Solve:
\[
\sqrt{x + 1} = \sqrt{x - 3} + 2
\]

*Solution:*
Isolate one square root:
\[
\sqrt{x + 1} - \sqrt{x - 3} = 2
\]
Square both sides:
\[
(\sqrt{x + 1} - \sqrt{x - 3})^2 = 2^2
\]
Expand the left-hand side:
\[
(x + 1) - 2\sqrt{(x + 1)(x - 3)} + (x - 3) = 4
\]
\[
2x - 2\sqrt{(x + 1)(x - 3)} - 2 = 4
\]
Isolate the remaining radical and square again to solve. (Details omitted for brevity, as this involves multiple steps.)

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#### *Example 3: Rational Equation Leading to a Radical*
Solve:
\[
\frac{1}{\sqrt{x}} = 2
\]

*Solution:*
Multiply through by \( \sqrt{x} \) to eliminate the denominator:
\[
1 = 2\sqrt{x}
\]
Isolate the square root:
\[
\sqrt{x} = \frac{1}{2}
\]
Square both sides:
\[
x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]
Check:
\[
\frac{1}{\sqrt{\frac{1}{4}}} = \frac{1}{\frac{1}{2}} = 2
\]
Valid solution.

Final answer:
\[
x = \frac{1}{4}
\]

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*Practice Problems*

1. Solve: \( \sqrt{x - 2} = 4 \)
2. Solve: \( \sqrt{2x + 3} - 1 = 5 \)
3. Solve: \( \sqrt{x} + 3 = x \)
4. Solve: \( \sqrt{4x + 1} = 3 \)
5. Solve: \( \frac{1}{\sqrt{x + 2}} = 1 \)

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*Key Skills Practiced*
Eliminating radicals using exponents.
Solving quadratic or linear equations that result from eliminating radicals.
Identifying and rejecting extraneous solutions.

This practice sharpens algebraic reasoning and builds foundational skills for advanced topics involving radicals.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths Please subscribe!    / nickperich  



I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study