Question

The equation \(\sqrt{x+1}-\sqrt{x-1}=\sqrt{4 x-1}\) has: (a) no solution (b) one solution (c) two solutions (d) more than two solutions

Short answer

" would be: The equation has (b) one solution.

Step-by-step solution

Rewrite the given equation

Write the equation as given: \(\sqrt{x+1}-\sqrt{x-1}=\sqrt{4 x-1}\).

Remove square roots by squaring both sides

To get rid of the square roots, square both sides of the equation: \((\sqrt{x+1}-\sqrt{x-1})^2=(\sqrt{4 x-1})^2\).

Simplify the equation

Expand the left-hand side of the equation and cancel out equal terms: \((x+1) -2\sqrt{(x+1)(x-1)}+(x-1) = 4x-1\).

Further simplify and rearrange the equation

Now, combine like terms: \(2x - 2\sqrt{(x+1)(x-1)}=4x-1-2x\). then we have, \(-2\sqrt{(x+1)(x-1)}=2x-1\).

Square both sides again

To remove the square root, combine the terms on both sides: \((-2\sqrt{(x+1)(x-1)})^2 = (2x-1)^2\).

Simplify and rearrange the equation

Expand the equation: \(4(x+1)(x-1)=4(x^2-1)\). Cancel out the 4 on both sides: \((x+1)(x-1)=x^2-1\).

Find possible solutions

Since the left side is already expanded, compare coefficients of both sides to find possible solutions: \(x^2-x-x+1 = x^2-1\). Rearrange and cancel: \(-2x+1=-1\). Hence, the possible solution is \(x=1\).

Check the solution for validity

Check the value of x by plugging it back into the original equation: \(\sqrt{(1)+1}-\sqrt{(1)-1}=\sqrt{4 (1)-1}\). This simplifies to: \(\sqrt{2}-\sqrt{0}=\sqrt{3}\). So the equation holds true, and the given value of x is valid. Therefore, the equation has #tag_answer#(b) one solution#tag_answer#.

Key concepts

Square Roots

Square roots are mathematical operations that are fundamental to solving radical equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.

When solving equations with square roots, such as the example provided where \(\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}\), the goal is to isolate the square roots and eventually remove them to simplify the equation. This is usually done by squaring both sides of the equation, because the square of a square root reverses the operation and leaves you with the original expression under the radical. However, care must be taken, especially when there are multiple square roots. Such cases often require the equation to be squared more than once to completely eliminate all square roots.

Squaring an equation can also introduce extraneous solutions—values that satisfy the squared form of the equation but not the original one—hence the importance of always checking the solutions in the original equation. In summary, understanding how to manipulate square roots is essential for solving radical equations.

Equation Simplification

Simplifying an equation is a process that makes solving it more manageable by combining like terms, expanding parentheses, and canceling out terms where possible. The step-by-step solution for the example radical equation illustrates this process.

After initially squaring both sides to eliminate the square roots, we simplify the equation by expanding the squared terms and then combining like terms, as seen in steps 3 and 4. The simplification usually involves recognizing and applying algebraic identities, such as \(a-b)^2 = a^2 - 2ab + b^2\) and \(a+b)(a-b) = a^2 - b^2\).

Further simplification comes from rearranging terms to isolate the remaining square root, leading to another squaring to eliminate the root entirely. The objective is to end up with a polynomial equation where the value of the variable in question can be clearly determined. Equation simplification is not only about making the equation easier to solve but also preparing it for the final step—validating solutions.

Validity Check in Equations

Once a solution is proposed for an equation, particularly one involving radical terms, it is crucial to conduct a validity check. This step ensures that our answer is not extraneous and that it actually satisfies the original equation without leading to any mathematical incorrectness, such as the square root of a negative number in the realm of real numbers.

In the given exercise, after finding the potential solution of \(x=1\), the validity check in step 8 is performed by substituting \(x\) back into the original equation. This confirms whether our solution is correct and adheres to the mathematical rules of square roots. The confirmation that \(\sqrt{2}-\sqrt{0}=\sqrt{3}\) is true completes the process, ensuring that our solution is indeed valid. This step is not merely a formality but a critical component of the problem-solving process, safeguarding against any missteps taken during the simplification and ensuring the accuracy of the result.