Problem 20 - Algebra 2 and Trigonometry

Question

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}+2 x+3} \\ {x+y=1}\end{array} $$

Step-by-Step Solution

Answer

The common solutions are \((-1, 2)\) and \((-2, 3)\).

Step 1: Express y in terms of x

Start by using the given equation \(x + y = 1\) to express \(y\) in terms of \(x\). Rearrange the equation to get \(y = 1 - x\).

Step 2: Substitute y in the quadratic equation

Replace \(y\) in the quadratic equation \(y = x^2 + 2x + 3\) with \(1 - x\) from Step 1. This gives the equation: \(1 - x = x^2 + 2x + 3\).

Step 3: Rearrange into standard quadratic form

Rearrange the equation from Step 2 to form a standard quadratic equation: \(x^2 + 3x + 2 = 0\) by moving all terms to one side.

Step 4: Factor the quadratic equation

Factor the quadratic \(x^2 + 3x + 2\) into \((x + 1)(x + 2) = 0\).

Step 5: Solve for x using the factored form

Set each factor equal to zero: \(x + 1 = 0\) and \(x + 2 = 0\). Solve for \(x\) to get \(x = -1\) and \(x = -2\).

Step 6: Find corresponding y values

Use the equation \(y = 1 - x\) to find \(y\) for each \(x\) value. For \(x = -1\), \(y = 1 - (-1) = 2\). For \(x = -2\), \(y = 1 - (-2) = 3\).

Step 7: Write the solutions as ordered pairs

The solutions to the system are \((-1, 2)\) and \((-2, 3)\).

Key Concepts

Quadratic Equations

Quadratic equations form the foundation for many algebraic concepts and are expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations describe parabolas when graphed, which can open upwards or downwards depending on the sign of \(a\). A quadratic equation can have different types of roots: real and distinct, real and repeated, or complex, depending on the discriminant \(b^2 - 4ac\). For instance, in the step-by-step solution provided, the quadratic equation \(x^2 + 3x + 2 = 0\) is used. It is essential to learn how to manipulate these equations, as they're pervasive in algebra and higher-level mathematics.

Algebraic Methods

Algebraic methods are systematic procedures that allow us to simplify and solve various types of equations, including systems of equations.

• **Substitution:** This involves solving one equation for one variable and then substituting that expression into another equation. In the provided problem, we first express \(y\) as \(1 - x\) and then substitute into the quadratic equation.
• **Elimination:** Another popular technique where you add or subtract entire equations from each other to eliminate one of the variables, often used in systems of linear equations.

These methods are powerful because they transform complex problems into simpler ones, allowing you to find solutions efficiently. Understanding these techniques is crucial for solving both single and systems of equations.

Factoring Quadratics

Factoring is a crucial skill in algebra, especially when working with quadratic equations. To factor a quadratic equation means to rewrite it as a product of its binomial factors.

• **Trial and Error:** Testing various factor pairs of the constant term until the correct combination is found.
• **Using Patterns:** Recognizing patterns such as perfect square trinomials or the difference of squares can expedite factoring.
• **Factoring by Grouping:** Useful for equations where simple factoring isn’t straightforward, involving grouping terms with a common factor and factoring them out.

In the step-by-step solution provided, we factored the quadratic \(x^2 + 3x + 2\) into \((x + 1)(x + 2)\). This allows us to solve for the roots easily. Mastery of factoring enables the solving of equation roots rapidly, and understanding these techniques can simplify many algebraic problems.

Ordered Pairs

In mathematics, an ordered pair is a pair of numbers describing a mathematical point in a two-dimensional space, usually denoted as (x, y).

• **Notation:** The first element of the pair, \(x\), is often called the abscissa, and the second element, \(y\), is called the ordinate.
• **Graphical Representation:** Ordered pairs are extensively used to locate points on a Cartesian plane.
• **Solution Representation:** When solving systems of equations, the solutions are expressed as ordered pairs, as these represent the point where the graphs of the equations intersect.

In the exercise at hand, after determining \(x = -1\) and \(x = -2\) through solved solutions, we associate respective \(y\) values, resulting in ordered pairs \((-1, 2)\) and \((-2, 3)\). These pairs graphically represent where both given equations share common solutions on a plane.