91Ó°ÊÓ

Factoring is a fundamental technique in solving quadratic equations. It involves rewriting the polynomial as a product of its factors. When you have a quadratic equation in the form of \(ax^2 + bx + c = 0\), the goal is to find two numbers that multiply to \(ac\) and add to \(b\). This step is crucial because it makes the equation easier to solve.

• Begin by identifying the constants \(a\), \(b\), and \(c\) from the equation.
• Search for two numbers that multiply to \(ac\) and sum to \(b\).
• Use these numbers to break down the middle term and factor by grouping.

In our solution, the equation \(x^2 + 5x + 6 = 0\) was factored into \((x+2)(x+3)=0\). This efficient method allows the complex quadratic equation to be expressed as a simple multiplication of linear terms, leading to straightforward solutions. Factoring is essential as it turns intimidating problems into manageable calculations.

Substitution plays a key role whenever you have multiple equations interlinked by common variables. In this exercise, we used substitution to transform expressions involving \(y\) back into terms of \(x\).

• Identify the equations given in the problem and see where substitution can simplify the scenario.
• Use where applicable to replace variables in one equation with terms from another.
• Transform the substituted equation into a recognizable form, such as a quadratic equation.

The step where we replaced \(y_1\) and \(y_2\) with \(x-3\) and \(x+8\) was pivotal in simplifying the problem to a quadratic equation: \((x - 3)(x + 8) = -30\). This technique reduces the complexity inherent in handling multiple expressions, focusing problem-solving efforts on a single equation in one variable.

Simplification is about making an equation or expression easier to work with by reducing its complexity. During simplification, each term of the equation is processed until a simpler, more easily solvable form is achieved.

• Perform arithmetic operations like distributing or combining like terms to clean up the equation.
• Move all terms to one side such that the equation becomes set to zero, often required for solving quadratic equations.
• Constantly check if further simplifications can be made at each step to ease the solution process.

In our example, simplifying was done by expanding the left side of the equation \((x - 3)(x + 8) = -30\) to get \(x^2 + 5x + 6 = 0\). This transformation prepares it for factoring, aligning it in the standard quadratic form, ready to be solved. Simplification is not just a process but a strategy for effective problem-solving.

Mathematical problem solving involves a systematic approach to tackling complex problems by breaking them down into simpler, more manageable tasks. This requires persistence and the practical application of various mathematical strategies and methods.

• Begin with analyzing given conditions and identify what is known and what must be found.
• Apply appropriate methods such as substitution and factoring to streamline and target the main problem-solving objective.
• Focus on each step logically, ensuring the solution proceeds in an organized manner towards finding the answer.

In our exercise, a mathematical problem-solving approach was followed. It started with the analysis of given conditions, moved to substitution and simplification, and concluded with factoring to find the solutions \(x = -2\) and \(x = -3\). Each step was a piece of the overall puzzle, guiding us to the solution. This approach nurtures not only the solution of the task at hand but also develops problem-solving skills for future applications.