Problem 12 - College Algebra Essentials

Question

Check all proposed solutions. $$ \sqrt{20-8 x}=x $$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{20-8x} = x \) is \(x = 2\)

Step 1: Isolate the square root

The given equation is \( \sqrt{20-8x} = x \). First, make sure that the square root is isolated.

Step 2: Square both sides

Squaring each side of the equation gets rid of the square root and allows us to simplify. In doing so, we get \( (20-8x) = x^2 \).

Step 3: Simplify and rewrite the equation

By further simplifying the equation and rewriting it as a quadratic, we get \(x^2 + 8x - 20 = 0 \).

Step 4: Solve the quadratic equation

We can solve this quadratic using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). By substituting \(a = 1\), \(b = 8\), \(c = -20\) into the formula, we get two solutions, \( x = 2 \) and \( x = -10\).

Step 5: Check the solutions

Substitute the two solutions back into the original equation. On doing so, we discover that only \(x = 2\) is a valid solution as \(x = -10\) yields \( \sqrt{20 - 8 * -10} = -10 \), which is not valid because a square root cannot equal a negative number

Key Concepts

Isolating the Square Root

When solving equations involving square roots, it is imperative to isolate the square root before proceeding with further steps. In the given example, we begin with the equation ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline {ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Squaring Both Sides

After isolating the square root on one side of the equation, the next step is to eliminate it by squaring both sides. This step is vital as it transforms the equation into one we can handle using algebraic methods we're more familiar with. In the provided exercise, when we square ewline ewline ewline ewline ewline ewline ewline ewline ewline

Quadratic Formula

The quadratic formula, ewline ewline ewline ewline ewline ewline ewline ewline ewline is a powerful tool for solving quadratic equations in the form ewline ewline ewline ewline ewline ewline ewline ewline ewline When we apply this formula to our simplified quadratic equation, we are able to find the potential solutions for ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Simplify Quadratic Equations

To solve a quadratic equation effectively, simplifying it into standard form is often a necessary step. The standard form is ewline ewline ewline ewline ewline ewline ewline ewline ewline This format makes it easier to identify the coefficients needed for applying the quadratic formula and simplifies the process of solving the equation. In the exercise, we rearranged the equation into ewline ewline ewline ewline ewline ewline ewline ewline ewline

Valid Solutions for Quadratics

Not all solutions obtained from the quadratic formula are necessarily valid in the context of the original problem. It's crucial to check each solution against the initial equation to ensure its validity. This step is sometimes referred to as the 'verification step'.In our example, when we plug the solutions back into the original equation, we discover that ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline because a square root cannot result in a negative value. This step underscores the importance of checking each solution within the context of the given equation. ewline ewline ewline ewline ewline ewline ewline ewline ewline