Solve each equation and verify your solution(s). Identify any extraneous roots. 2√6 + 5𝑥 − 2 = 3𝑥
To solve the equation, we first need to combine like terms. We have:
2√6 + 5𝑥 − 2 = 3𝑥
Next, let's subtract 5𝑥 from both sides to isolate the 𝑥 term:
2√6 - 2 = 3𝑥 - 5𝑥
Combining like terms:
2√6 - 2 = -2𝑥
Now, let's divide both sides by -2 to solve for 𝑥:
(2√6 - 2) / -2 = -2𝑥 / -2
Simplifying:
-√6 + 1 = 𝑥
So the solution to the equation is 𝑥 = -√6 + 1.
To verify this solution, substitute 𝑥 = -√6 + 1 back into the original equation:
2√6 + 5(-√6 + 1) - 2 = 3(-√6 + 1)
Simplifying:
2√6 - 5√6 + 5 - 2 = -3√6 + 3
-3√6 + 3 = -3√6 + 3
The equation is verified. There are no extraneous roots in this case.
To solve the equation 2√6 + 5𝑥 − 2 = 3𝑥, we can follow these steps:
Step 1: Move the constant term to the other side of the equation.
2√6 + 5𝑥 − 2 − 5𝑥 = 3𝑥 − 5𝑥
Simplifying this gives:
2√6 - 2 = -2𝑥
Step 2: Simplify the expression on the left side.
2√6 - 2 = -2𝑥
Step 3: Move the coefficient of 𝑥 to the other side of the equation.
-2 = -2𝑥
Step 4: Solve for 𝑥.
Dividing through by -2 gives us:
𝑥 = 1
Step 5: Verify the solution.
Substitute 𝑥 = 1 back into the original equation:
2√6 + 5(1) − 2 = 3(1)
2√6 + 5 − 2 = 3
Simplifying:
2√6 + 3 = 3
This equation is not true, which means 𝑥 = 1 is not a valid solution.
Therefore, the original equation 2√6 + 5𝑥 − 2 = 3𝑥 has no valid solutions.
To solve the equation 2√6 + 5𝑥 − 2 = 3𝑥, we'll follow these steps:
Step 1: Isolate the radical term.
To do this, subtract 3𝑥 from both sides of the equation:
2√6 + 5𝑥 − 3𝑥 − 2 = 0
Simplifying, we have:
2√6 + 2𝑥 − 2 = 0
Step 2: Move the constant term to the other side.
To do this, add 2 to both sides of the equation:
2√6 + 2𝑥 − 2 + 2 = 0 + 2
Simplifying, we have:
2√6 + 2𝑥 = 2
Step 3: Isolate the variable term.
To do this, subtract 2√6 from both sides of the equation:
2√6 + 2𝑥 − 2√6 = 2 - 2√6
Simplifying, we have:
2𝑥 = 2 - 2√6
Step 4: Solve for 𝑥.
To solve for 𝑥, we divide both sides of the equation by 2:
(2𝑥)/2 = (2 - 2√6)/2
Simplifying, we have:
𝑥 = 1 - √6
So the solution to the equation is 𝑥 = 1 - √6.
Now let's verify this solution.
Verify:
Substitute the value of 𝑥 we found, which is 1 - √6, back into the original equation:
2√6 + 5(1 - √6) − 2 = 3(1 - √6)
Simplifying the equation, we have:
2√6 + 5 - 5√6 − 2 = 3 - 3√6
Combining like terms, we get:
3 - 3√6 = 3 - 3√6
The equation is true, so 𝑥 = 1 - √6 is a valid solution.
Now let's check for any extraneous roots. An extraneous root can occur when a value satisfies the simplified form of the equation but not the original equation.
In this case, we have not performed any operations that introduce extraneous roots, such as squaring both sides or dividing by a variable term. Therefore, there are no extraneous roots in this equation.
So 𝑥 = 1 - √6 is the only solution to the equation 2√6 + 5𝑥 − 2 = 3𝑥.