Question

Solve for \(x\) and check. $$\sqrt{x^{2}-7.25}=8.75-x$$

Short answer

The solution to the equation is approximately \(x = 4.79\).

Step-by-step solution

Square both sides

To eliminate the square root, square both sides of the equation: \((\sqrt{x^{2}-7.25})^{2} = (8.75-x)^{2}\). This results in the equation \(x^{2} - 7.25 = 76.5625 - 17.5x + x^{2}\).

Simplify the equation

Since \(x^{2}\) appears on both sides of the equation, they cancel each other out. This simplifies the equation to \(-7.25 = 76.5625 - 17.5x\).

Rearrange the equation

To isolate the variable \(x\), add \(17.5x\) and \(7.25\) to both sides of the equation, which gives you \(17.5x = 76.5625 + 7.25\).

Combine like terms

Combine the numbers on the right side to get \(17.5x = 83.8125\).

Solve for x

Divide both sides by \(17.5\) to solve for \(x\), which gives you \(x = \frac{83.8125}{17.5} = 4.78928571429\). Rounded to two decimal places, \(x = 4.79\).

Check the solutions

Substitute \(x = 4.79\) into the original equation to check if both sides are equal: \(\sqrt{(4.79)^{2} - 7.25} \stackrel{?}{=} 8.75 - 4.79\). Calculate both sides to confirm that the left-hand side equals the right-hand side.

Key concepts

Square Root Equations

Understanding how to solve square root equations is an essential skill in algebra. A square root equation is an equation containing a variable within a square root. To solve such equations, the primary goal is to first isolate the square root on one side of the equation.

Using the given exercise \(\sqrt{x^{2}-7.25}=8.75-x\), we start by focusing on the square root and ensuring that it is the only term on one side. In more complex cases, you may need to perform additional steps, such as moving terms or factoring, to achieve this isolation. Once the square root stands alone, you square both sides to eliminate the square root. This is a critical step that must be handled with care because when you square a square root, it effectively removes the root, leaving you with just the radicand—the expression inside the root.

Isolating the Variable

Once a square root has been removed, the next phase involves isolating the variable. This means manipulating the equation in such a way that the variable is by itself on one side, which is key to finding the equation's solution. In our exercise, after removing the square root, we get \(x^{2}-7.25=76.5625-17.5x+x^{2}\).

The steps to isolate the variable may include combining like terms, such as subtracting or adding the same term to both sides, or using inverse operations. When we rearrange the equation properly and combine like terms, the variable \(x\) is alone on one side. This allows us to solve for \(x\) by performing appropriate arithmetic operations, such as addition or division.

Simplifying Equations

Simplifying the equation is a pivotal step that makes the problem more manageable and prepares us for the final solution. Let's take our example where we were left with the equation \(x^{2}-7.25=76.5625-17.5x+x^{2}\), and we see that \(x^{2}\) terms appear on both sides. Simplifying involves several techniques, including:

• Canceling equal terms on both sides of the equation,
• Combining like terms to reduce the equation to fewer terms,
• Factoring expressions if applicable,
• Reducing fractions to their simplest form.

After simplifying as much as possible, we are often left with a more straightforward equation that is easier to solve.

Checking Solutions

Once we have found a potential solution for our equation, such as \(x=4.79\) from the prior steps, it is crucial to verify its accuracy by checking the solution. Plugging the value back into the original equation helps ensure that our operations did not introduce any errors.

In our exercise, we check by substituting \(x\) with \(4.79\) into the original square root equation, and simplifying both sides. If both sides of the equation have equal values, our solution is correct. It is worth noting that sometimes, especially when squaring both sides, we may end up with extraneous solutions that don't satisfy the original equation. Thus, checking is an imperative step that must always be carried out to confirm the validity of the solution. Remember, the correctness of our solutions is just as important as our ability to solve the equation itself.