Problem 45 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$x=\sqrt{4 x+45}$$

Step-by-Step Solution

Answer

The solution set for the equation \(x=\sqrt{4 x+45}\) is {9} after checking for extraneous solutions.

Step 1: Isolate the square root

The equation is already in the form where the square root is isolatable, meaning that \(x = \sqrt{4x + 45}\). The square root is already isolated on the right side.

Step 2: Eliminate the square root

Remove the square root by squaring both sides of the equation: \(x^2 = (4x + 45)\). We get \(x^2 = 4x + 45\).

Step 3: Solve the quadratic equation

Rearrange the equation to equal zero: \(x^2 - 4x - 45 = 0\). Next, factor the quadratic, which will give: \((x - 9)(x + 5) = 0\). Setting each factor equal to zero yields two potential solutions: \(x = 9\) and \(x = -5\).

Step 4: Check for extraneous solutions

We must ensure that these are actual solutions. We place them back into the original equation: \(\sqrt{4x + 45}\). When we place \(x = 9\) into the equation we get 9 = \(\sqrt{4(9) + 45}\). Simplifying gives 9 = \(\sqrt{81}\), which means 9 = 9. This is true, so \(x = 9\) is a valid solution. When we place \(x = -5\) into the equation, however, the result is \(-5 = \sqrt{4(-5) + 45}\), which simplifies to \(-5 = \sqrt{-5}\). Because the square root of a negative number is not a real number, \(x = -5\) is an extraneous solution and should be dismissed.

Key Concepts

Extraneous Solutions

When solving equations, particularly those involving operations like square roots, you may encounter solutions that don't actually work in the original equation. These are what we call "extraneous solutions." They often arise because of the way the math manipulations are carried out, such as squaring both sides of an equation.

Here’s what to do to manage these solutions effectively:

• Always double-check: Insert your solution back into the original equation to verify its validity.
• Be cautious with transformations: Certain operations, like squaring, can introduce solutions that aren’t valid.
• Look out for domain restrictions: Equations involving \sqrt{...} have domain restrictions since we typically deal with real numbers.

For example, in the equation \(x = \sqrt{4x + 45}\), checking the solutions helps identify \(x = -5\) as not feasible within the original equation’s constraints because it results in the square root of a negative number.

Factoring Quadratic Equations

Factoring is a key method to solve quadratic equations, which often appear in various mathematical problems. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\). There's a simple process to follow that can help make factoring easier.

Here's a quick guide:

• Set the equation to zero: First, you want to move all terms to one side so the equation reads \(x^2 + bx + c = 0\).
• Look for two numbers: These numbers should multiply to \(c\) and add up to \(b\).
• Break it down: Once you find these numbers, factor the quadratic into two binomials.

For example, the equation \(x^2 - 4x - 45 = 0\) can be factored into \((x - 9)(x + 5) = 0\). This allows us to then solve for \(x\), finding potential solutions that need to be verified later.

Isolating Square Roots

Isolating a square root in an equation is a necessary step when trying to solve it. By getting the square root term by itself on one side of the equation, you're in a position to eliminate it, usually by squaring both sides. This is useful especially in equations like \(x = \sqrt{4x + 45}\), where you can work to simplify and ultimately solve the problem.

• Identify the root: Make sure the square root is alone on one side, as seen in this equation.
• Eliminate the root: Squaring both sides helps to remove the square root, transforming the equation into a quadratic equation.
• Don't forget about constraints: Remember, operations like squaring can change the equation subtly, so it's critical to double check against the original form to ensure no errors are introduced.

The importance of isolating and then removing the square root is crucial in order to move ahead with solving the equation. This technique is a fundamental part of solving many algebraic problems where square roots are present.