Problem 119 - College Algebra

Question

Describe two methods for solving this equation: \(x-5 \sqrt{x}+4-0\)

Step-by-Step Solution

Answer

For Method 1, after rearranging and squaring the equation to eliminate the square root, the quadratic equation \(x^2 - 17x + 16 = 0\) is obtained. It can be solved to find the solutions for x. In Method 2, by substituting \( \sqrt{x} \) with a new variable y, we get a quadratic equation in terms of y - \(y^2 - 5y + 4 = 0\). Then, y can be solved and substituted back into \( \sqrt{x} \) to find the x values.

Step 1: Method 1 - Rearranging and Squaring

First, rearrange the equation by moving \( \sqrt{x} \) to the other side:\(x + 4 = 5 \sqrt{x} \)Now, square both sides of the equation to get rid of the square root:\( (x + 4)^2 = (5 \sqrt{x})^2 \)Simplify to obtain a quadratic equation:\( x^2 + 8x + 16 = 25x \)Move all terms to one side to obtain:\(x^2 - 17x + 16 = 0 \)Solve the quadratic equation (either by factoring, completing the square or using the quadratic formula) to find the solutions for x.

Step 2: Method 2 - Substitution Method

This method involves substituting \( \sqrt{x} \) with a new variable. Let's call the new variable y. So, \( y = \sqrt{x} \) or \( x = y^2 \)Substitute y into the equation:\(y^2 - 5y + 4 = 0\)This is a quadratic equation, solve for y, then substitute y back into \( \sqrt{x} \) to find the solutions for x.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations graph as parabolas when plotted on a coordinate plane. The general solution involves finding values of \(x\) that satisfy the equation, and there are several methods to do so. Understanding the structure of a quadratic equation is essential, as it helps determine the best method for solving it. Quadratic equations can be solved by:

• Factoring: Rewriting the quadratic equation as a product of two binomials.
• Completing the Square: Transforming the equation into a perfect square trinomial.
• Quadratic Formula: A formula that gives the roots directly as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Choosing the right method often depends on the specific problem and the numbers involved.

Squaring Method

The squaring method is a useful technique when dealing with equations containing a square root, such as \(x - 5\sqrt{x} + 4 = 0\). The goal is to eliminate the square root by squaring both sides of the equation. However, care must be taken, as squaring can introduce extraneous solutions, which are solutions that do not satisfy the original equation.

To apply the squaring method:

• Rearrange the equation: Isolate the square root on one side, for instance, \(x + 4 = 5\sqrt{x}\).
• Square both sides: Convert to \((x+4)^2 = (5\sqrt{x})^2\), simplifying to \(x^2 + 8x + 16 = 25x\).
• Solve the resulting quadratic: Move all terms to one side: \(x^2 - 17x + 16 = 0\).

This converts the problem into a more familiar quadratic equation that can be solved using conventional methods.

Substitution Method

The substitution method is another technique for solving equations, particularly useful when an equation involves awkward terms like a square root. In this method, substitution is used to transform the original equation into a simpler form.

Here's how the substitution method works for the given problem:

• Identify the substitution: Use \(y = \sqrt{x}\), thus \(x = y^2\) to replace \(\sqrt{x}\) in the equation.
• Substitute and simplify: Change \(x - 5\sqrt{x} + 4 = 0\) into \(y^2 - 5y + 4 = 0\).
• Solve the simpler quadratic: Solve for \(y\) using the quadratic equation techniques.
• Back-substitute to find \(x\): Replace \(y\) with \(\sqrt{x}\) to find the values of \(x\).

This approach reduces complexity by allowing you to work with a standard quadratic equation.