Radical Roots NotesRadical Roots Topic: Solving Equations with One Radical NAME: DATE: The Mission Today we are learning to "undo" square roots. If we want to solve for \(x\), we have to peel away the layers. What is the opposite of a square root? Key Vocabulary Radical Equation: An equation where the variable is found ______________________________________________. Inverse Operation: Squaring a square root _________________ the radical. The Golden Rule Whatever you do to one side, you MUST _________________________. The Blueprint: 4 Steps to Success 1 ISOLATE Get the radical ___________ on one side of the equal sign. 2 SQUARE Square ________ sides to eliminate the radical. 3 SOLVE Solve the remaining equation for the ________________. 4 CHECK Plug your answer back into the ____________ equation! Guided Examples Example A: Level 1 \[\sqrt{x + 5} = 9\] The Check: Example B: Level 2 \[2\sqrt{x} - 4 = 10\] The Check: The Job Site: Individual Practice 01 \[\sqrt{3x - 2} = 5\] Step-by-Step Workspace 02 \[\sqrt{x + 10} + 3 = 8\] Step-by-Step Workspace 03 \[2\sqrt{x + 4} = 10\] Remember Step 1! Step-by-Step Workspace Quality Control Check Why is it absolutely critical to check your solutions when solving radical equations? Describe in your own words below.
Radical Roots KeyRadical Roots | ANSWER KEY Topic: Solving Equations with One Radical TEACHER REFERENCE Key Vocabulary Radical Equation: An equation where the variable is found underneath the radical symbol (radicand). Inverse Operation: Squaring a square root cancels out or eliminates the radical. The Golden Rule Whatever you do to one side, you MUST DO TO THE OTHER SIDE. The Blueprint: 4 Steps to Success ISOLATE alone SQUARE both SOLVE variable CHECK original Guided Examples Example A: Level 1 \[\sqrt{x + 5} = 9\] \[(\sqrt{x + 5})^2 = 9^2\] \[x + 5 = 81\] \[x = 76\] The Check: \[\sqrt{76 + 5} = \sqrt{81} = 9 \checkmark\] Example B: Level 2 \[2\sqrt{x} - 4 = 10\] \[2\sqrt{x} = 14\] \[\sqrt{x} = 7\] \[(\sqrt{x})^2 = 7^2\] \[x = 49\] The Check: \[2\sqrt{49} - 4 = 2(7) - 4 = 14 - 4 = 10 \checkmark\] Practice Answer Key 01 \[\sqrt{3x - 2} = 5\] \[3x - 2 = 25\] \[3x = 27\] \[x = 9\] Check: \(\sqrt{3(9)-2} = \sqrt{25} = 5 \checkmark\) 02 \[\sqrt{x + 10} + 3 = 8\] \[\sqrt{x + 10} = 5\] \[x + 10 = 25\] \[x = 15\] Check: \(\sqrt{15+10}+3 = 5+3=8 \checkmark\) 03 \[2\sqrt{x + 4} = 10\] \[\sqrt{x + 4} = 5\] \[x + 4 = 25\] \[x = 21\] Check: \(2\sqrt{21+4} = 2\sqrt{25} = 2(5) = 10 \checkmark\) Quality Control Check (Key) It is essential to check our work to ensure our solution actually solves the original radical expression. This verifies that no arithmetic errors occurred and that the solution is valid for the square root's domain.
Radical Roots SlidesRadical Roots Solving Single-Radical Equations Blueprint No. 001 The Mission "Our goal is to isolate the unknown variable trapped inside a radical." Inverse Power To undo a square root (\(\sqrt{}\)), we use the power of 2 (squaring). \( (\sqrt{x})^2 = x \) The Golden Rule of Inverses The 4-Step Blueprint 1 ISOLATE Get the radical by itself on one side. 2 SQUARE Square both sides to kill the root. 3 SOLVE Solve the resulting equation. 4 CHECK Verify your answer for errors! Example A: The Basics Difficulty: Level 1 \[\sqrt{x + 5} = 9\] 1 Radical is already isolated! \(\checkmark\) 2 Square both: \(x + 5 = 81\) 3 Solve: \(x = 76\) 4 Check: \(\sqrt{76 + 5} = \sqrt{81} = 9\) Example B: Isolation Difficulty: Level 2 \[2\sqrt{x} - 4 = 10\] 1 Isolate first! \(2\sqrt{x} = 14 \rightarrow \sqrt{x} = 7\) 2 Square: \(x = 49\) 3 Solved! \(x = 49\) 4 Check: \(2\sqrt{49} - 4 = 14 - 4 = 10 \checkmark\) The Job Site PROBLEM 01 \[\sqrt{3x - 2} = 5\] PROBLEM 02 \[\sqrt{x + 10} + 3 = 8\] PROBLEM 03 \[2\sqrt{x + 4} = 10\] Grab your guided notes and start building!
