Problem 57 - Introductory Algebra for College Students

Question

Solve: \(\sqrt{2 x+3}=2 x-3 .\) (Section 8.5, Example 4)

Step-by-Step Solution

Answer

The equation \(\sqrt{2 x+3}=2 x-3\) has no valid solution for real numbers.

Step 1: Isolate the square root

To isolate the square root, we first rearrange the equation, so the square root is by itself on one side of the equation: \(\sqrt{2 x+3}=2 x-3\).

Step 2: Deal with the square root

We then eliminate the square root by squaring both sides of the equation: \((\sqrt{2 x+3})^{2}=(2x-3)^{2}\). This gives: \(2x + 3 = 4x^{2} - 12x + 9\).

Step 3: Create a quadratic equation

We can now rearrange the equation into standard quadratic form: \(0 = 4x^{2} - 12x + 6\).

Step 4: Solve the quadratic equation

This is a quadratic equation in the form of \(ax^{2}+bx+c=0\). We usually solve it using the quadratic formula \(x=[-b±\sqrt{((b^{2})-4ac)}/(2a)\]. But this equation doesn't yield any real solutions.

Step 5: Check your solution

Since there's no real solution to the quadratic equation, we need to check that our initial manipulation didn't introduce extraneous solutions. When we test the original equation \(\sqrt{2 x+3}=2 x-3\) with our solution, we find that the square root function doesn't have a solution in the real number domain which fits the equations, and thus the original equation doesn't have a real solution either.

Key Concepts

Solving Equations

When it comes to solving equations, the goal is to find the value of the variable that makes the equation true. Here are key steps to follow:

• Identify the variable you need to solve for. In our exercise, that's x.
• Isolate the term with the variable on one side of the equation. This often involves rearranging or simplifying the equation.
• Apply operations like addition, subtraction, multiplication, or division, as necessary, to both sides of the equation to keep it balanced. These operations help move other numbers to the opposite side.
• Focus on simplifying the equation step-by-step until you reach the variable by itself with its corresponding numerical value.

In some cases, like with quadratic equations, you might get complex expressions that require special methods, such as factoring, completing the square, or using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \] to find the solutions.

Square Roots

Square roots offer a unique challenge when solving equations. A square root, symbolized as \( \sqrt{} \), is a number that, when multiplied by itself, returns the original number under the square root.

• When an equation involves a square root, your first step is often to isolate this square root expression.
• Then eliminate the square root by squaring both sides of the equation. This step turns the original expression into a newer expression without the square root.

It's vital to remember that squaring both sides can sometimes introduce additional solutions (more on that later). This manipulation can often lead to a quadratic equation.

Extraneous Solutions

Extraneous solutions are surprising results! These solutions may appear to fit the transformed equation but do not satisfy the original problem.

• They often arise when transformations, like squaring both sides, are performed.
• Always substitute your solutions back into the original equation to check their validity.
• In our exercise, when solving the transformed equation, we concluded there were no real x values that satisfy both the original and transformed equations.

Recognizing and discarding extraneous solutions is crucial, as they can lead to incorrect conclusions if not properly identified and eliminated. Testing them with the original equation will determine if they hold true.