Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$(3 x-2)^{2}=(x-5)(9 x+4)$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{24}{29}\).

Step 1: Expand the Left Side

To expand the left side of the equation \[(3x - 2)^2,\]use the formula for the square of a binomial: \[(a-b)^2 = a^2 - 2ab + b^2.\]Here, \(a = 3x\) and \(b = 2\). Expanding gives:\[(3x)^2 - 2(3x)(2) + (2)^2 = 9x^2 - 12x + 4.\]

Step 2: Expand the Right Side

Expand the right side\[(x-5)(9x+4)\]by using the distributive property: \[(x)(9x) + (x)(4) + (-5)(9x) + (-5)(4).\]Carrying out these multiplications results in:\[9x^2 + 4x - 45x - 20.\]Simplifying this expression gives:\[9x^2 - 41x - 20.\]

Step 3: Set the Expanded Equations Equal to Each Other

Equate the expanded left side and right side expressions:\[9x^2 - 12x + 4 = 9x^2 - 41x - 20.\]

Step 4: Simplify and Solve for x

Subtract \(9x^2\) from both sides to remove \(x^2\) terms, resulting in:\[-12x + 4 = -41x - 20.\]Add \(41x\) to both sides to move the \(x\) terms to the left:\[29x + 4 = -20.\]Subtract 4 from both sides to isolate the term with \(x\):\[29x = -24.\]Finally, divide by 29 to solve for \(x\):\[x = -\frac{24}{29}.\]

Key Concepts

Solving Quadratic Equations

Quadratic equations are like delicious puzzles waiting to be solved! Imagine they are in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Sometimes, they might disguise themselves with other numbers or letters. But fear not! You can tackle these equations using different methods. The quadratic formula, factoring, or by completing the square - each is a tool in your math toolkit. In the original problem, after dealing with expansions, the equations simplify nicely, and you subtract similar terms to pinpoint the solution. Here, isolating \(x\) leads us to the answer \(x = -\frac{24}{29}\). When you approach any quadratic equation, think about how it can be rearranged to match the form and solution you're working toward. Each step takes you closer to that satisfying conclusion!

Expansion of Binomials

To expand binomials like \((3x - 2)^2\), we use the binomial expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\). It's like opening up a box and seeing all the parts inside! Here, \(a = 3x\) and \(b = 2\). Once expanded, the expression becomes \(9x^2 - 12x + 4\). This method helps you break down expressions into their components, making complex-looking problems much simpler. By practicing the expansion of binomials, you gain skills in handling different expressions, opening the doors to further mathematical explorations. Don't forget, each term in the expanded form comes from interacting the original parts with each other. It's a dance of math magic!

Distributive Property

When you apply the distributive property, it’s like spreading the math love all around! It helps you multiply a single term over a sum or difference. In the equation \((x-5)(9x+4)\), you apply the distributive property to expand it: \((x)(9x) + (x)(4) + (-5)(9x) + (-5)(4)\).

• The first step is to multiply everything by \(x\), giving you \(9x^2 + 4x\).
• Next, do the same with \(-5\), resulting in \(-45x - 20\).

Add these products together, and you get the expanded form \(9x^2 - 41x - 20\). The distributive property is a foundational skill in algebra and helps manage the flow of numbers and variables across expressions. Remember, it's like sharing each part across multiple layers to reveal the full picture of the expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means making them as neat and tidy as possible. After expanding both sides of the equation, we encountered terms like \(9x^2 - 12x + 4 = 9x^2 - 41x - 20\). To simplify, subtract \(9x^2\) from both sides. This erases the \(x^2\) terms. Then, organize like terms together:

• Add \(41x\) to \(-12x\), resulting in \(29x\).
• Move constants together by subtracting 4 from both sides, yielding \(29x = -24\).

Finally, solve for \(x\) by dividing both sides by 29, and you've got \(x = -\frac{24}{29}\)! Simplifying helps untangle the knot of equations, revealing the simple, elegant truth behind the math. Always remember, clarity in your math work leads to clarity in your solutions!