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Draw a line from each equation in Column A to its solution in Column B.
Column A Column B
1. 3x 1 5 5 24 x 5 3
2. 4x2 5 16 x 5 22 or x 5 2
3. 6x 2 2 5 16 x 5 23
Vocabulary Builder
extraneous (adjective) ek stray nee us
Definition: Extraneous means not essential or relevant to the situation or subject being considered.
Math Usage: An extraneous solution is one that comes from the solving process but is not a valid solution of the original equation.
Main Idea: A radical equation is an equation that has a variable in a radicand. When solving radical equations, it is possible to obtain extraneous solutions. You must check each solution in the original equation.
Use Your Vocabulary
If you solve the equation 2!x 5 x, you get two solutions: 0 and 1. The check for each solution is shown below.
Then I divide each side by 2 to isolate the radical.
Now I square each side.
Next I simplify.
Finally, I multiply each side by 3.3.
= 2√ 𝓵𝓵3.3
= √ 𝓵𝓵3.3
= 𝓵𝓵3.3
= 𝓵𝓵
First I substitute 1 for t in t = 2√ 𝓵𝓵3.3 .
= 2
303 Lesson 10-4
Solving by Isolating the Radical
Got It? What is the solution of !x 2 5 5 22?
5. Circle the first step to solve the equation !x 2 5 5 22. Underline the second step.
Add 5 to both sides. Square both sides. Subtract 22 from both sides.
6. Complete the steps to solve 7. Check your solution by substituting !x 2 5 5 22. for x in the original equation.
5 3 Å 2 5 0 22
x 5 2 5 0 22
0 22
8. The solution I found in Exercise 6 does / does not solve the equation.
Using a Radical Equation
Got It? The time t in seconds it takes for a pendulum of a clock to complete a
full swing is approximated by the equation t 5 2Å<
3.3, where < is the length of the pendulum in feet. If the pendulum of a clock completes a full swing in 1 s, what is the length of the pendulum? Round to the nearest tenth of a foot.
9. Circle what you are trying to find to solve the problem.
length of the pendulum time it takes for 1 swing time it takes for 2 swings
10. Complete the reasoning model below to solve the equation.
11. To the nearest tenth of a foot, the pendulum is ft long.
Got It? What is the solution of !7x 2 4 5 !5x 1 10?
12. The equation is solved below. Write a justification for each step.
(!7x 2 4)25 (!5x 1 10)
2
7x 2 4 5 5x 1 10
2x 2 4 5 10
2x 5 14
x 5 7
13. Check your solution.
When you solve an equation by squaring each side, you write a new equation. The new equation may have solutions that do not make the original equation true.
Original Square New Solutions of Equation each side. Equation New Equation
x 5 3 x2 5 32 x2 5 9 3, 23
In the example above, 23 is not a solution of the original equation. It is an extraneous solution.
Identifying Extraneous Solutions
Got It? What is the solution of 2y 5 !y 1 6?
14. Solve the equation.
y2 5 Square each side.
5 0 Subtract y 1 6 from each side.
(y 2 )(y 1 ) 5 0 Factor the quadratic equation.
y 2 5 0 or y 1 5 0 Use the Zero-Product Property.
y 5 or y 5 Solve for y.
15. Check to see whether each result satisfies the original equation.
23 0 Å 1 6 2 0 Å 1 6
23 0 Å 2 0 Å
23 2 2 5
16. The only solution of the equation 2y 5 !y 1 6 is .