Solving Radical Equations
Solving Radical Equations walks Grade 6 students through a three-step process: isolate the radical, square both sides to eliminate it, then solve and check for extraneous solutions. This topic is covered in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots. Checking answers is critical because squaring both sides can introduce false solutions that do not satisfy the original equation.
Key Concepts
Property To Solve a Radical Equation 1. Isolate the radical on one side of the equation. 2. Square both sides of the equation. 3. Continue as usual to solve for the variable.
The technique of squaring both sides may introduce extraneous solutions , which are solutions that do not work in the original equation. Therefore, you must check your final answers.
Examples To solve $\sqrt{x} = 8$, square both sides: $(\sqrt{x})^2 = 8^2$, which gives $x = 64$. The check, $\sqrt{64}=8$, works. To solve $\sqrt{y 2} + 5 = 9$, first isolate the radical to get $\sqrt{y 2} = 4$. Squaring both sides gives $y 2 = 16$, so $y = 18$. To solve $\sqrt{z} = 6$, squaring both sides gives $z = 36$. However, checking this in the original equation gives $\sqrt{36} = 6 \neq 6$. Therefore, there is no solution.
Common Questions
How do you solve a radical equation?
First isolate the radical on one side, then square both sides to remove the radical, solve the resulting equation, and always check your answers in the original equation.
What are extraneous solutions in radical equations?
Extraneous solutions are values that satisfy the squared equation but not the original radical equation. They are introduced when you square both sides.
Why do you check solutions in radical equations?
Squaring both sides can create false solutions. Substituting your answer back into the original equation confirms it is a valid solution.
Where is solving radical equations in Yoshiwara Elementary Algebra?
It is in Chapter 9: More About Exponents and Roots of Yoshiwara Elementary Algebra.
What if there is a radical on both sides of the equation?
Isolate one radical, square both sides, and if another radical remains, isolate and square again. Always check the final answer.