Problem 33 - Algebra 2 and Trigonometry

Question

Write a quadratic equation with integer coefficients for each pair of roots. \(-2,-1\)

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + 3x + 2 = 0\).

Step 1: Define the roots

We know the roots of the quadratic equation are given as \(-2\) and \(-1\). These roots can be represented using \(x = -2\) and \(x = -1\).

Step 2: Write the equation using the roots

The standard form of a quadratic equation with roots \(r_1\) and \(r_2\) is \((x - r_1)(x - r_2) = 0\).Plug in the given roots: \((x - (-2))(x - (-1)) = 0\),which simplifies to \((x + 2)(x + 1) = 0\).

Step 3: Expand the expression

Now expand the expression \((x + 2)(x + 1)\):You'll perform the distribution: \(x(x + 1) + 2(x + 1)\),this results in \(x^2 + x + 2x + 2\).

Step 4: Combine like terms

Combine the terms from the expansion: \(x^2 + x + 2x + 2\) simplifies to \(x^2 + 3x + 2\). This represents the quadratic equation with integer coefficients.

Key Concepts

Integer Coefficients

In a quadratic equation, the goal is often to have integer coefficients. These are just whole numbers, including negatives. Utilizing integer coefficients makes equations simpler and more relatable when solving real-life problems.

With roots given as \(-2\) and \(-1\), we can form a quadratic expression with integer coefficients by following specific mathematical rules. In the equation \(ax^2 + bx + c = 0\), all terms \(a\), \(b\), and \(c\) should be integers, ensuring our equation remains simple to handle.

For example, with the roots \(-2\) and \(-1\), when expanded, this yields the equation \(x^2 + 3x + 2 = 0\), maintaining integer coefficients throughout the process. By keeping coefficients as integers, we simplify both the conceptual understanding and computational execution of algebraic tasks.

Roots of Equations

The roots of an equation are the solutions where the equation equals zero. In simpler terms, they are the values of \(x\) for which the equation \(ax^2 + bx + c = 0\) holds true. They play a crucial role in formulating the equation itself.

For our exercise, the roots given are \(-2\) and \(-1\). This means if you substitute either of these numbers back into the equation, the left side will balance out to zero:

• At \(-2\): \[x^2 + 3x + 2 = 0 \, simplifies \, to \, ((-2)^2 + 3(-2) + 2 = 0.)\]
• At \(-1\): \[x^2 + 3x + 2 = 0 \, simplifies \, to \, ((-1)^2 + 3(-1) + 2 = 0.)\]

Recognizing these roots allows engineers or scientists to backtrack and deduce the quadratic equation if they know the outcomes from specific inputs.

Polynomial Expansion

Polynomial expansion is the method of simplifying expressions involving products of polynomials. It can seem complex, but with repetition, it becomes much more intuitive. For quadratic equations with known roots, the standard format is \((x - r_1)(x - r_2) = 0\).

Expanding these products through distribution is as follows:

We have, \( (x + 2)(x + 1) = 0\). To expand this:

• First distribute \(x\): \(x(x+1) = x^2 + x\)
• Then distribute \(+2\): \(2(x+1) = 2x + 2\)

By combining these like terms, \(x^2 + x + 2x + 2\) simplifies further into \(x^2 + 3x + 2\). Understanding this helps not just in mathematics, but in physics or even fields like climate modeling or financial forecasts, where relationships between variables need to be expressed compactly.