Problem 43 - Algebra 2 and Trigonometry

Question

Solve and check each equation. \(2 a+\sqrt{50}=\sqrt{98}\)

Step-by-Step Solution

Answer

The solution is \(a = \sqrt{2}\).

Step 1: Isolate the variable term

First, identify and isolate the variable term on one side of the equation. We have the equation \(2a + \sqrt{50} = \sqrt{98}\). To isolate \(2a\), subtract \(\sqrt{50}\) from both sides of the equation.

Step 2: Calculate the square roots

Calculate the square roots on both sides. \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\) and \(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\). Substitute these values into the equation to get: \(2a + 5\sqrt{2} = 7\sqrt{2}\).

Step 3: Solve for the variable

Subtract \(5\sqrt{2}\) from \(7\sqrt{2}\) to get: \(2a = 2\sqrt{2}\). Then, divide both sides by 2 to solve for \(a\): \(a = \sqrt{2}\).

Step 4: Verify the solution

Substitute \(a = \sqrt{2}\) back into the original equation to verify. Compute \(2a = 2\sqrt{2}\), so the left side becomes \(2\sqrt{2} + 5\sqrt{2} = 7\sqrt{2}\), which matches \(\sqrt{98}\), confirming that the solution is correct.

Key Concepts

Isolating Variables

Solving equations often starts with isolating the variable, which means getting the variable by itself on one side of the equation. This helps in finding its value easily. In the given example, the equation involves the term \(2a\). To isolate this variable, we need to move any extra terms away from it. The equation given is \(2a + \sqrt{50} = \sqrt{98}\).
To isolate \(2a\), we have to remove \(\sqrt{50}\) from the left side of the equation. This is done by subtracting \(\sqrt{50}\) from both sides. Here's what it looks like:

• Original Equation: \(2a + \sqrt{50} = \sqrt{98}\)
• Subtract \(\sqrt{50}\): \(2a = \sqrt{98} - \sqrt{50}\)

Now, \(2a\) is isolated, making it easier to solve for \(a\). Isolating variables is a fundamental step in solving equations, as it simplifies the problem and paves the way for further calculations.

Calculating Square Roots

With \(2a\) isolated, the next step is understanding the numbers involved, particularly those under square roots. The task here is to simplify both \(\sqrt{50}\) and \(\sqrt{98}\). Breaking down these square roots helps in solving the equation more smoothly.
Here's how square roots are calculated step by step:

• \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)
• \(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\)

Both values are simplified in terms of \(\sqrt{2}\), which is helpful because it allows a clean subtraction later on.
Substituting these simplified values back into the isolated equation changes it to:

• \(2a = 7\sqrt{2} - 5\sqrt{2}\)
• \(2a = 2\sqrt{2}\)

Simplifying square roots by breaking them down into their prime factors makes the math much less intimidating and more manageable to handle.

Verifying Solutions

The last and crucial step in solving equations is to verify the solution. This means checking that your obtained answer indeed satisfies the original equation. For our equation, we found that \(a = \sqrt{2}\). Now, let’s see if substituting this value back into the original equation works.
Perform these checks:

• Calculate \(2a\) with \(a = \sqrt{2}\): \(2a = 2\sqrt{2}\)
• Substitute back into the equation: \(2\sqrt{2} + 5\sqrt{2} = 7\sqrt{2}\)
• The right-hand side of the original equation is \(\sqrt{98}\), which simplifies to \(7\sqrt{2}\)

Since both sides equal \(7\sqrt{2}\), the solution \(a = \sqrt{2}\) is confirmed correct.
Verification is a small but effective step to ensure no mistakes were made, and that the solution aligns perfectly with the original problem, wrapping up the solving process with certainty.