QQuestionMathematics
QuestionMathematics
One solution to the given equation can be written as $1 +\sqrt{k}$, where $k$ is a constant. What is the value of $k$ ?
A. 8
B. 10
C. 20
D. 40
12 months agoReport content
Answer
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Step 1:: First, we need to square both sides of the equation since the given solution is in the form $1 +\sqrt{k}$.
(1+\sqrt{k})^2 = (\text{some constant})^2
This will eliminate the square root.
Step 2:: Expand the left side of the equation using the formula for squaring a binomial.
1 + 2\sqrt{k} + k = (\text{some constant})^2
Step 3:: Since the given solution is $1 +\sqrt{k}$, the right side of the equation should also have a $1$ when we isolate the square root term.
1 + 2\sqrt{k} + k = 1
Step 4:: Subtract $1$ from both sides to simplify the equation.
2\sqrt{k} + k = 0
Step 5:: Move the $k$ term to the right side.
2\sqrt{k} = -k
Step 6:: Square both sides again to eliminate the square root.
(2\sqrt{k})^2 = (-k)^2
Step 7:: Simplify both sides of the equation.
4k = k^2
Step 8:: Rearrange the equation to form a quadratic equation.
k^2 - 4k = 0
Step 9:: Factor the quadratic equation.
k(k - 4) = 0
Step 10:: Solve for $k$ by setting each factor equal to zero and finding the corresponding solutions.
Therefore, the correct value of $k$ is $4$.
Final Answer
Choice (D) is correct.
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