Problem 84 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: }(x+3)^{2}=(x+3)(x-5)+7 $$

Step-by-Step Solution

Answer

x = -17/8

Step 1: Expand the equation

First, expand both sides of the equation. For the left side, use the formula for a binomial square: \( (a+b)^2 = a^2 + 2ab + b^2 \) with \( a = x \) and \( b = 3 \). This gives: \( (x+3)^2 = x^2 + 6x + 9 \).For the right side, distribute \( (x+3) \): \( (x+3)(x-5) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \).So the original equation becomes:\( x^2 + 6x + 9 = x^2 - 2x - 15 + 7 \).

Step 2: Simplify the equation

Combine the terms on the right side of the equation. \( x^2 - 2x - 15 + 7 = x^2 - 2x - 8 \).This simplifies our equation to:\( x^2 + 6x + 9 = x^2 - 2x - 8 \).

Step 3: Eliminate common terms

Subtract \( x^2 \) from both sides of the equation to eliminate the \( x^2 \) term from both sides:\( x^2 + 6x + 9 - x^2 = x^2 - 2x - 8 - x^2 \). This simplifies to:\( 6x + 9 = -2x - 8 \).

Step 4: Combine like terms

Add \( 2x \) to both sides to combine like terms:\( 6x + 2x + 9 = -2x + 2x - 8 \). This simplifies to:\( 8x + 9 = -8 \).

Step 5: Isolate the variable

Subtract 9 from both sides to isolate the variable term:\( 8x = -8 - 9 \).This simplifies to:\( 8x = -17 \).

Step 6: Solve for x

Divide both sides by 8 to solve for \( x \):\( x = \frac{-17}{8} \).

Key Concepts

binomial expansion

The binomial expansion is a crucial concept in algebra, as it allows us to expand expressions that involve sums raised to a power. One common formula is for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).In our exercise, we're given \((x + 3)^2\), so applying the formula, where \(a = x\) and \(b = 3\), we get:\(x^2 + 2 \times x \times 3 + 3^2 = x^2 + 6x + 9\).This step is essential because it simplifies the left-hand side of the equation, making it easier to perform subsequent algebraic manipulations.Remember:

• Each term in the expansion comes from multiplying terms from the binomial.
• The coefficients of each term can be found using combinations or by applying the formula directly.

equation simplification

Simplifying equations is a foundational skill in algebra, making complex equations more manageable. In our problem, once we expanded both sides, we had to combine like terms to move towards a simpler form. The initial equation was:\(x^2 + 6x + 9 = x^2 - 2x - 15 + 7\).We simplified this by combining like terms on the right side:\(x^2 - 2x - 15 + 7 = x^2 - 2x - 8\).This process dramatically reduces the equation's complexity, turning it into:\(x^2 + 6x + 9 = x^2 - 2x - 8\).Tips for simplifying equations:

• Always combine like terms, which are terms with the same variables raised to the same power.
• Be consistent with the operations you perform on both sides of the equation.

isolating variables

Isolating the variable is the technique used to get the variable (e.g., \(x\)) on one side of the equation. Our goal is to find the value of \(x\). Starting from:\(6x + 9 = -2x - 8\),we perform a series of operations:

• Add \(2x\) to both sides: \(6x + 2x + 9 = -2x + 2x - 8\), simplifying to \(8x + 9 = -8\).
• Subtract 9 from both sides: \(8x = -8 - 9\), simplifying to \(8x = -17\).
• Finally, divide both sides by 8: \(x = \frac{-17}{8}\).

Key points to remember while isolating variables:

• Perform the same operation on both sides of the equation to maintain equality.
• Focus on isolating the variable term by step-by-step simplification.
• Move all terms involving the variable to one side and constant terms to the other.

solving quadratic equations

Solving quadratic equations, such as \(ax^2 + bx + c = 0\), usually involves several methods: factoring, using the quadratic formula, or completing the square. However, in our exercise, the equation simplifies in such a way that it behaves more like a linear equation by eliminating the \(x^2\) terms. The step-by-step approach involved expanding, simplifying, and isolating terms to eventually solve for \(x\).
When working with standard quadratics, you might use:

• Factoring: Find a pair of numbers that multiply to \(ac\) and add to \(b\).
• Quadratic formula: Use \(x = \frac{-b \text{±} \frac{\text{√(b^2 - 4ac)}}{2a}\).
• Completing the square: Manipulate the equation into \((x + p)^2 = q\) form.

In our exercise, the quadratic nature was simplified through steps eliminating the quadratic component, demonstrating that not all quadratic equations require traditional techniques.
Essential takeaways:

• Always look for opportunities to simplify the equation first.
• Understand when an equation can be reduced to a simpler form.
• Follow logical steps systematically to solve for the variable.