More Involved Radical Equation Example | Summary and Q&A

TL;DR

Learn how to solve radical equations step by step by isolating radicals and squaring both sides of the equation.

Key Insights

• 🙃 Isolate one of the radicals by subtracting it from both sides of the equation.
• 🙃 Squaring both sides of the equation eliminates the radicals and simplifies the expression.
• 😑 Use the formula a + b^2 = a^2 + 2ab + b^2 to expand and simplify the squared expression.
• ❓ Validate the solutions by substituting them back into the original equation.

Transcript

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Questions & Answers

Q: How do you solve radical equations step by step?

To solve radical equations, first isolate one of the radicals by subtracting it from both sides of the equation. Then, square both sides to eliminate the radicals. Continue simplifying the equation until you have a quadratic equation, and use the quadratic formula to find the solutions for x. Validate the solutions by substituting them back into the original equation.

Q: What is the importance of squaring both sides of the equation?

Squaring both sides of the equation eliminates the radicals and allows us to simplify the expression. It also helps us transform the equation into a quadratic form, making it easier to find the solutions for x using the quadratic formula.

Q: Why is it necessary to validate the solutions?

Validating the solutions is necessary because squaring both sides of the equation can introduce extraneous solutions. These solutions are not valid solutions to the original equation and result from squaring the equation multiple times. By substituting the solutions back into the original equation, we can determine if they are valid or extraneous.

Q: What is an extraneous solution?

An extraneous solution is a solution that does not satisfy the original equation. It can occur when squaring both sides of the equation introduces additional solutions that are not valid. To identify extraneous solutions, substitute the solutions back into the original equation and check if they satisfy it.

Summary & Key Takeaways

• To solve radical equations, isolate one of the radicals by subtracting it from both sides of the equation.

• Square both sides of the equation to eliminate the radicals and simplify the expression.

• Expand and simplify the squared expression using the formula a + b^2 = a^2 + 2ab + b^2.

• Continue simplifying the equation by performing operations such as addition, subtraction, and multiplication.

• Reduce the equation to a quadratic equation by collecting like terms.

• Use the quadratic formula to find the solutions for x.

• Validate the solutions by plugging them back into the original equation.

• Identify any extraneous solutions, which are solutions that result from squaring the equation multiple times.