Problem 22 - Algebra A Combined Function

Question

Solve. $$ \sqrt{2 x-4}-\sqrt{3 x+4}=-2 $$

Step-by-Step Solution

Answer

The solutions are \( x = 4 \) and \( x = 20 \).

Step 1: Isolate One Square Root

Rewrite the equation to isolate one of the square roots. Add \( \sqrt{3x + 4} \) to both sides: \( \sqrt{2x - 4} = \sqrt{3x + 4} - 2 \).

Step 2: Square Both Sides

Square both sides to eliminate the square root. This gives: \( (\sqrt{2x - 4})^2 = (\sqrt{3x + 4} - 2)^2 \), which simplifies to \( 2x - 4 = (3x + 4) - 4 \sqrt{3x + 4} + 4 \).

Step 3: Simplify the Equation

Combine like terms: \( 2x - 4 = 3x + 8 - 4 \sqrt{3x + 4} \). Rearrange to isolate the square root term: \( 4 \sqrt{3x + 4} = 3x + 8 - 2x + 4 \). Simplify: \( 4 \sqrt{3x + 4} = x + 12 \).

Step 4: Isolate Square Root Again

Divide both sides by 4 to solve for the square root: \( \sqrt{3x + 4} = \frac{x + 12}{4} \).

Step 5: Square Both Sides Again

Square both sides once more to eliminate the square root again: \( (\sqrt{3x + 4})^2 = \left( \frac{x + 12}{4} \right)^2 \), which gives \( 3x + 4 = \frac{(x + 12)^2}{16} \).

Step 6: Clear the Fraction

Multiply both sides by 16 to clear the fraction: \( 16(3x + 4) = (x + 12)^2 \). This simplifies to \( 48x + 64 = x^2 + 24x + 144 \).

Step 7: Rearrange into a Quadratic Equation

Rearrange the equation into standard quadratic form: \( x^2 - 24x + 80 = 0 \).

Step 8: Solve the Quadratic Equation

Factor the quadratic equation: \( (x - 20)(x - 4) = 0 \). This gives potential solutions for \( x \) as \( x = 20 \) and \( x = 4 \).

Step 9: Check Solutions

Substitute \( x = 20 \) and \( x = 4 \) back into the original equation to verify. For \( x = 20 \), after simplifying, \( \sqrt{36} - \sqrt{64} = 6 - 8 = -2 \), which is true. For \( x = 4 \), \( \sqrt{4} - \sqrt{16} = 2 - 4 = -2 \), which also holds true.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. These equations are typically in the form \( ax^2 + bx + c = 0 \). The distinguishing feature of a quadratic equation is the highest exponent of the variable being 2, making it a second-degree polynomial.
Quadratics can appear in various forms and solving them often requires different techniques, including factoring, completing the square, or using the quadratic formula.

• In our problem, isolating the square root terms eventually leads to a quadratic equation. This is common when solving square root equations, as the squaring process can create terms with higher exponents.
• Once in the form \( x^2 - 24x + 80 = 0 \), we're able to employ specific techniques to solve for \( x \).

Understanding the characteristics of quadratic equations sets the groundwork for using specific solving techniques efficiently. The solutions to a quadratic equation can be complex, real, or repeated, all of which carry important implications depending on the context of the problem.

Factoring Techniques

Factoring is a mathematical process of breaking down a complex expression into simpler components. For quadratic equations, it involves finding two binomial expressions that multiply to give the original quadratic expression.
When an equation like \( x^2 - 24x + 80 = 0 \) is presented, factoring helps in identifying potential solutions for \( x \) by expressing the quadratic in the form \( (x - p)(x - q) = 0 \).

• In our exercise, we factor the equation into \( (x - 20)(x - 4) = 0 \). This step reveals the roots of the equation as \( x = 20 \) and \( x = 4 \).
• A correct factorization is necessary to ensure all potential solutions are explored. It's always good to verify your factors by expanding them back to the original quadratic form.

Factoring not only helps find solutions but also provides insight into the nature of the equation, such as the axis of symmetry and vertex in the case of quadratic graphs.

Algebraic Manipulation

Algebraic manipulation is essential in solving equations, as it involves rearranging, simplifying, and combining terms to isolate the variable of interest. This process is critical in equations involving square roots, as seen in our problem.
Initially, manipulating the equation \( \sqrt{2x-4}-\sqrt{3x+4}=-2 \) by isolating and then squaring terms allows us to simplify the equation sequentially.

• Rewriting one side to simplify, such as \( \sqrt{2x - 4} = \sqrt{3x + 4} - 2 \), helps manage the complexity involved with radical expressions.
• Each step refines the equation, eventually leading to a simplified form suitable for further steps like factoring or using a quadratic formula.

Mastering algebraic manipulation not only aids in solving specific equations but also builds the skill set necessary for addressing more complex algebraic problems across various mathematical areas.