Problem 61 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. $$ A=2 x^{2}+4 x h ; \quad \text { for } x $$

Step-by-Step Solution

Answer

\( x = -h \pm \frac{1}{2}\sqrt{8A + 16h^2} \)

Step 1: Arrange the equation

Start with the given equation: \( A = 2x^2 + 4xh \). Our goal is to solve for \( x \). Notice that the equation is quadratic in form.

Step 2: Set the equation to zero

Rearrange the equation to fit the standard quadratic form \( ax^2 + bx + c = 0 \). Subtract \( A \) from both sides to get \( 2x^2 + 4xh - A = 0 \).

Step 3: Identify parameters for quadratic formula

In a quadratic equation \( ax^2 + bx + c = 0 \), identify: \( a = 2 \), \( b = 4h \), and \( c = -A \).

Step 4: Use the quadratic formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used to find \( x \). Substitute the identified values: \( x = \frac{-4h \pm \sqrt{(4h)^2 - 4 \cdot 2 \cdot (-A)}}{2 \cdot 2} \).

Step 5: Simplify the expression

Calculate inside the square root: \( b^2 - 4ac = (4h)^2 + 8A = 16h^2 + 8A \). Substitute back: \( x = \frac{-4h \pm \sqrt{16h^2 + 8A}}{4} \).

Step 6: Further simplify

Simplify the expression by dividing each term in the numerator by 4: \( x = -h \pm \frac{\sqrt{16h^2 + 8A}}{4} \). Finally, distribute the division: \( x = -h \pm \frac{1}{2}\sqrt{8A + 16h^2} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the heart of solving equations, especially quadratic ones. It involves rearranging an equation to isolate a variable you are solving for. The goal is to make your target variable the subject. If it's tangled up in terms, algebraic manipulation helps you untangle it efficiently. To perform algebraic manipulation:

• Identify like terms and constants.
• Use operations like addition or subtraction to move terms across the equation.
• Factorize or expand expressions if needed.

In our exercise, we start with the equation: \[ A = 2x^2 + 4xh \]To target the variable \( x \), it's essential to rearrange the equation to form a recognizable quadratic expression, which we achieve by setting the equation to zero. This is a key step in making sure you can apply methods like the quadratic formula later on.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula is derived from the process of completing the square, allowing you to find the values of the variable \( x \) that satisfy the equation.The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]With this formula, you can calculate solutions for \( x \) by simply plugging in the coefficients \( a \), \( b \), and \( c \) extracted from your equation. It's especially helpful because it accounts for complex roots when the expression inside the square root (the discriminant) is negative.In our example, we identify \( a = 2 \), \( b = 4h \), and \( c = -A \) from the quadratic equation obtained after rearranging the original equation. By substituting these values into the quadratic formula, we can solve for \( x \), setting the stage for further simplification of the expression.

Solving Equations

Solving equations generally requires a systematic approach. When dealing with quadratics, your strategy includes preparing the equation, applying the quadratic formula, and simplifying the result where possible.Here’s how this applies to our case:1. **Prepare the Equation**: Get the equation into the form \( ax^2 + bx + c = 0 \), allowing the use of the quadratic formula.2. **Apply the Quadratic Formula**: Substitute the coefficients to find the potential values of \( x \).3. **Simplify the Expression**: After applying the formula, you may need to perform additional simplification.In our specific example, once we have used the quadratic formula:\[ x = \frac{-4h \pm \sqrt{16h^2 + 8A}}{4} \]Further simplification leads us to:\[ x = -h \pm \frac{1}{2}\sqrt{8A + 16h^2} \]This final step shows how algebra and mathematical logic work together to provide a clean, interpretable solution.