Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ x=3+\sqrt{5 x-9} $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Isolate the square root

To start solving the equation, isolate the square root on one side. The given equation is already in the form \( x = 3 + \sqrt{5x - 9} \). The square root is isolated on the right side of the equation.

Step 2: Square both sides

To eliminate the square root, square both sides of the equation. Start with the given equation: \[ x = 3 + \sqrt{5x - 9} \]Square both sides to get:\[ x^2 = (3 + \sqrt{5x - 9})^2 \] Expanding the right side gives:\[ x^2 = 9 + 6\sqrt{5x - 9} + (5x - 9) \] This simplifies to:\[ x^2 = 5x + 6\sqrt{5x - 9} \].

Step 3: Eliminate the square root

Rearrange the equation to isolate the term containing the square root:\[ x^2 - 5x = 6\sqrt{5x - 9} \]. Then, divide both sides by 6 to get the square root isolated:\[ \frac{x^2 - 5x}{6} = \sqrt{5x - 9} \].

Step 4: Square both sides again

To eliminate the square root completely, square both sides again:\[ \left( \frac{x^2 - 5x}{6} \right)^2 = 5x - 9 \]. Expand and simplify the left side:\[ \frac{(x^2 - 5x)^2}{36} = 5x - 9 \].

Step 5: Simplify the equation

Multiply both sides by 36 to remove the fraction:\[ (x^2 - 5x)^2 = 36(5x - 9) \]. Simplify both sides:\[ (x^2 - 5x)^2 = 180x - 324 \].

Step 6: Solve the resulting polynomial equation

Expand the polynomial:\[ (x^2 - 5x)^2 = x^4 - 10x^3 + 25x^2 \].Equate it to the simplified right side:\[ x^4 - 10x^3 + 25x^2 = 180x - 324 \].Rearrange terms to form a polynomial equation:\[ x^4 - 10x^3 + 25x^2 - 180x + 324 = 0 \].

Step 7: Solve for \( x \)

Solve the polynomial equation using techniques such as factoring, synthetic division, or numerical methods. The solutions to this equation are the potential values for \( x \). By testing, find that \( x = 4 \) works and no other real solutions satisfy the original equation.

Key Concepts

Square Root Isolation

In algebra, square root isolation is a strategy to solve equations involving square roots. By isolating the square root, we make it easier to eliminate the radical and solve the equation completely.
The given exercise was:

• \( x = 3 + \sqrt{5x - 9} \)

The square root \( \sqrt{5x - 9} \) is already isolated on one side of the equation. This means it's all set for the next steps, like squaring both sides of the equation, without needing extra rearrangement.
By focusing first on isolating the square root, you minimize errors and streamline the solution process. It also sets a clear path for what we'll do next—removing that pesky square root by squaring both sides.

Polynomial Expansion

Once the square root is isolated, the next step often involves eliminating it, as seen in our exercise by squaring both sides of the equation.

• Start with \( x^2 = (3 + \sqrt{5x - 9})^2 \)
• After squaring, apply the foil method: \( x^2 = 9 + 6\sqrt{5x - 9} + (5x - 9) \)

Expanding a polynomial requires applying algebraic rules and operations to multiply terms and simplify expressions. This converts complicated terms into a recognizable polynomial form. Understanding how to expand polynomials involves breaking down each term and methodically applying multiplication to every pair of terms involved. This process fully opens up the expression, making any terms that need further simplification or manipulation more apparent. When dealing with radicals, this expansion reveals any leftover square roots, thus signaling that further work is necessary to fully resolve the equation.

Polynomial Equation

A polynomial equation is one that can be expressed in the standard form of a polynomial. The main goal in polynomial equations is to simplify them into a form that can be solved for its roots.
By squaring both sides again in our exercise, we reach:

• \( \left( \frac{x^2 - 5x}{6} \right)^2 = 5x - 9 \)

This process ultimately leads us to a polynomial equation:

• \( x^4 - 10x^3 + 25x^2 - 180x + 324 = 0 \)

Writing a polynomial equation often requires moving all terms to one side of the equation for a nice clean zero on the other side. That gives you a chance to work with standard polynomial-solving techniques like factoring. Whether you're working with quadratic, cubic, or higher-degree polynomials, the strategy remains essentially the same: simplify and either factor or solve using appropriate methods, always starting from isolating the terms as needed.

Factoring Techniques

Solving polynomial equations often involves using factoring techniques, essential in breaking down an equation into a product of simpler factors. This makes it easier to find the roots (solutions) of the equation.
In the exercise, having the equation

• \( x^4 - 10x^3 + 25x^2 - 180x + 324 = 0 \)

requires looking for patterns or applying methods like synthetic division to find solutions. For more complex polynomials, sometimes trial and error or numerical methods are used to test potential roots. In this case, testing revealed that \( x = 4 \) satisfies the equation.
Once a root is found, it can help simplify the equation further or confirm the solution. Factoring is about recognizing structures in the polynomial that can be simplified, usually by reversing polynomial multiplication processes or by trial and test-based approaches for discovering specific roots.