Problem 23 - College Algebra Essentials

Question

Check all proposed solutions. $$ \sqrt{x+8}-\sqrt{x-4}=2 $$

Step-by-Step Solution

Answer

Analyzing the solutions we got, we see that \(x=-6\) should be excluded because it would lead to a complex number as a result in the original equation. Therefore, the solution to the equation is \(x= 18\)

Step 1: Isolate one radical

First rearrange the equation to isolate one of the radicals on one side of the equation, which in this case can be \(\sqrt{x-4}\). It gives: \(\sqrt{x-4} = \sqrt{x+8} - 2\).

Step 2: Square both sides

Square each side of the equation to remove the square root. This gives: \((\sqrt{x-4})^2 = (\sqrt{x+8} - 2)^2\)On simplifying, we have \(x - 4 = (x+8) - 4\sqrt{x+8} + 4\).

Step 3: Rearrange to isolate the other radical

As the equation still contains a radical sign, rearrange the equation to isolate terms involving the square root (which is \(\sqrt{x+8}\)). So, you get:\(4\sqrt{x+8} = x + 4\).

Step 4: Square both sides again

Now, square both sides of the equation to remove the remaining square root. This gives:\(16(x+8) = (x+4)^2\).On simplifying, we get \(16x + 128 = x^2 + 8x + 16\).

Step 5: Simplify the equation

Transform the equation into a classic quadratic equation by subtracting \(16x + 128\) from both sides. This results in:\(x^2 - 8x - 112 = 0\).

Step 6: Solve the quadratic equation

Finally, solve the quadratic equation for \(x\) using the quadratic formula. The solutions are: \(x = -6\) and \(x = 18'\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations in the form \(ax^2 + bx + c = 0\). This formula can turn seemingly complex equations into easily solvable problems. It provides the solutions (or roots) for the variable \(x\). Importantly, the quadratic formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]Here are the steps to utilize the quadratic formula:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Plug these values into the formula.
• Compute the discriminant \(b^2 - 4ac\) to determine the nature and number of solutions.
• Calculate the two potential values for \(x\) using "+" and "-" in the formula.

In the given exercise, after isolating and squaring the radicals, we ended up with a quadratic equation \(x^2 - 8x - 112 = 0\). By applying the quadratic formula, we found the solutions \(x = -6\) and \(x = 18\). This confirmed the calculation and provided the possible solutions to the initial radical equation.

Radical Expressions

Radical expressions contain a radical symbol (√), typically involving the square root. They often appear less intimidating if you break them down into manageable parts. Solving equations with radicals involves specific steps:

• Isolate the radical expression on one side of the equation to simplify the right-hand side as much as possible.
• Find a way to eliminate the radical, usually by squaring both sides of the equation.
• Simplify the resulting equation, which might lead to a linear or quadratic equation.

In the exercise, we encounter radical expressions \(\sqrt{x+8}\) and \(\sqrt{x-4}\). A key step was to reposition one radical to facilitate the squaring process. By eliminating radicals through squaring, we simplified to more traditional algebraic expressions, paving the way for solving the equation. Remember that squaring both sides may introduce extraneous solutions, so it's essential to verify each result against the original equation.

Squaring Both Sides

Squaring both sides of an equation is a typical strategy to eliminate radicals. This powerful approach bares all when an equation involves square roots (radicals).Steps for squaring both sides:

• Make sure to isolate a radical expression on one side of the equation as much as possible.
• Convert the equation to quadratic form by squaring both sides.
• Simplify the resulting equation.

In the original exercise, we squared both sides twice to systematically remove all radicals. After isolating \(\sqrt{x-4}\), we squared both sides to eliminate the radical. Doing so revealed a new equation that included another radical expression, \(\sqrt{x+8}\). We then repeated the strategy and squared both sides again. Each squaring action broadened our equation, eventually leading us to a form solvable via the quadratic formula.Remember, squaring can ease the solution of equations involving radicals, but always check for extraneous solutions that don't satisfy the original equation. These often appear due to the nature of squaring, which can turn a negative into a positive, thus introducing non-factorable solutions.