Problem 17 - Algebra and Trigonometry

Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $$x-\sqrt{2 x+5}=5$$

Step-by-Step Solution

Answer

The solution for the given equation is \(x = 2\). However, initially two solutions were obtained \(x = 2\) and \(x = 10\), but the latter was discarded after checking, because it rendered the original equation false.

Step 1: Isolate the Square Root

Rearrange the terms to isolate the square root on one side of the equation: \( x - 5 = \sqrt{2x + 5} \).

Step 2: Square Both Sides

Squaring removes the radical, transforming the equation: \( (x - 5)^{2}=(\sqrt{2x+5})^{2} \). Simplify to obtain \(x^{2} -10x + 25 = 2x + 5\).

Step 3: Simplify and Solve the Quadratic Equation

Reorder to standard quadratic form \(x^{2} - 12x + 20 = 0\). Find the solutions by using the quadratic formula \(x = [-(-12) \pm \sqrt{(-12)^{2} - 4*1*20}]/(2*1)\). The solutions for x are \(x_{1,2} = 12 \pm \sqrt{144 - 80} \) / 2, so \(x_{1,2} = 6 \pm \sqrt{64}/2\), becoming \(x_{1,2} = 6 \pm 8/2\). Therefore, \(x_1 = 2\) and \(x_2 = 10\).

Step 4: Check the Solutions

Substitute both solutions back into the original equation to verify whether any are extraneous. For \(x_1 = 2\), it is valid as \(2 - \sqrt{2*2+5} = 5 - 3 = 2\) which is a true statement. For \(x_2 = 10\), also by substituting this solution it occurs \(10 - \sqrt{2*10 + 5} = 10 - \sqrt{25} = 10 - 5 = 5\) which is not a true statement since it should be 10. So, \(x = 10\) is an extraneous solution and will be discarded.

Key Concepts

Isolate the Square Root

When solving radical equations, it's essential to begin by isolating the square root on one side of the equation. This step simplifies the process of eliminating the radical, allowing us to work with a more straightforward algebraic expression. Take the example of the equation \(x - \sqrt{2x + 5} = 5\). Here, we move all terms without a radical to the other side to get \(x - 5 = \sqrt{2x + 5}\).

Isolating the square root can sometimes entail additional steps depending on the complexity of the equation. Key objectives in this phase include being cautious not to disturb the equality and preparing the equation for squaring, which will eliminate the square root. Always perform inverse operations carefully to maintain the balance of the equation.

Quadratic Formula

After isolating and subsequently squaring the square root, our problem often transforms into a quadratic equation. The quadratic formula \(x = [-b \pm \sqrt{b^{2} - 4ac}]/(2a)\) provides the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\).

For instance, if we derive the quadratic equation \(x^2 - 12x + 20 = 0\), we can determine the solutions by plugging the coefficients a, b, and c into the quadratic formula. The real beauty of this formula lies in its ability to consistently find the solutions of any quadratic equation regardless of the complexity, even when factoring is not straightforward or possible.

Check Extraneous Solutions

An essential part of solving radical equations is to verify all potential solutions, as some might be extraneous, meaning they do not satisfy the original equation. These solutions can emerge from the process of squaring both sides since squaring is not a one-to-one function but rather a many-to-one function.

For the solution \(x_1 = 2\) and \(x_2 = 10\), we substitute them back into the initial equation to ensure they hold true. Through careful substitution, we discover that \(x_2 = 10\) does not satisfy the original equation, indicating it's an extraneous solution. It's crucial to always perform this check as the algebraic manipulation can introduce these false solutions.

Quadratic Equations

Quadratic equations are fundamental elements of algebra that take the form \(ax^2 + bx + c = 0\) where a, b, and c are coefficients, and 'a' is not equal to zero. They represent parabolas when graphed on a coordinate plane and are distinguished by having a degree of two—the highest power of 'x' is squared.

Regardless of the method used to solve them, whether it's factoring, completing the square, or employing the quadratic formula, understanding quadratic equations is crucial for tackling a variety of mathematical problems. They frequently appear in various applications, including physics, engineering, economics, and countless other fields that utilize mathematical modeling.