Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation algebraically. Check your solutions. $$\sqrt{x}-\sqrt{x-5}=1$$

Step-by-Step Solution

Answer

The solution to the given equation is \( x = \frac{29}{4} \).

Step 1: Isolate a Square Root

Firstly, we need to isolate one of the square roots. We can start with \( \sqrt{x-5} \). So, move \(\sqrt{x-5}\) to the other side of the equation which becomes: \( \sqrt{x} = 1+\sqrt{x-5} \)

Step 2: Square Both Sides

Next step is to square both sides of the equation to remove the square root. \( (\sqrt{x})^2 = (1+\sqrt{x-5})^2 \) which simplifies to \( x = 1 + 2*\sqrt{x-5} + (x-5) \). Hence, we get \( 0 = 1 + 2*\sqrt{x-5} - 4 = 2*\sqrt{x-5}-3\).

Step 3: Isolate Second Square Root

Isolate the remaining square root: \( \sqrt{x-5} = \frac{3}{2} \).

Step 4: Square Again

Square both sides of the equation to remove the square root entirely. \( (x-5) = \left(\frac{3}{2}\right)^2 \). So, \( x = 5 + \left(\frac{3}{2}\right)^2 \).

Step 5: Solve for x

Find the value of x: \( x= 5 + \frac{9}{4} = \frac{20}{4} + \frac{9}{4} = \frac{29}{4} \).

Step 6: Check Solution

Finally, substitute the calculated x into the original equation to validate the solution. So \(\sqrt{\frac{29}{4}}-\sqrt{\frac{29}{4}-5} = 1\), it checks out as the LHS equals to the RHS of the original equation, making \( x = \frac{29}{4} \) valid

Key Concepts

Square roots

Square roots are an essential concept in algebra and many other areas of mathematics. They are used to find a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9.

When dealing with equations involving square roots, like the one in our original exercise,

• Start by understanding that the square root symbol (\(\sqrt{\cdot} \)) implies taking the positive number that results from squaring.
• Recognize that square roots can complicate solving equations since they are non-linear and can involve both real and imaginary numbers.
• To eliminate a square root, squaring the number is a common technique, reversing the operation of taking a square root by doing the inverse operation.

This sets the stage for isolating variables, which is often the first step after understanding the components of the equation you're dealing with.

Isolating variables

Isolating variables is crucial for simplifying an equation to make solving it possible. It means rearranging the equation so that you're left with the variable on one side of the equation.

In our example, we first aimed to isolate the square root \(\sqrt{x-5}\) by moving it across the equation. This helps focus on dealing with one square root at a time and simplifies the expression markedly.

• Move terms around using simple algebraic operations such as addition, subtraction, multiplication, or division.
• It’s generally easier to deal with one square root at a time to prevent the equation from becoming overly complex.

Remember, isolating variables is a stepping stone to clearer and more manageable equations, setting the groundwork for squaring equations and checking solutions.

Squaring equations

Squaring equations is a useful tool to eliminate square roots, allowing the equation to be simplified to a more conventional algebraic form. This technique is especially handy when dealing with square roots on one or both sides of an equation.

In our exercise, once we isolated the square root, the next logical step was to square the entire equation to clear it:

• Carefully square both sides of the equation simultaneously; remember each element inside the parenthesis gets squared too.
• Be mindful of the distribution; when squaring a binomial, remember to apply the formula \((a+b)^2 = a^2 + 2ab + b^2\).

This tactic transforms the equation into a polynomial, a format that is generally easier to handle and solve, leading us towards finding explicit solutions.

Checking solutions

After you’ve solved an equation, checking your solution is a crucial step to confirm its correctness. This step involves substituting your solution back into the original equation to ensure that it holds true.

In the original example, we found \(x = \frac{29}{4}\). To verify this

• Insert \(x\) back into the equation: \(\sqrt{\frac{29}{4}} - \sqrt{\frac{29}{4} - 5}\).
• Compute each component to ensure both sides of the equation are equal.

Checking solutions not only assures correctness but also highlights any potential errors in calculations or logical steps previously overlooked. This validation step is always recommended to maintain accuracy in problem-solving.