Problem 30 - Algebra 2 and Trigonometry

Question

Write a quadratic equation with integer coefficients for each pair of roots. \(2,5\)

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 - 7x + 10 = 0\).

Step 1: Identify the Roots

The roots given are 2 and 5. In a quadratic equation, these are the values of \(x\) that satisfy the equation.

Step 2: Write the Equation Form with Roots

A quadratic equation with roots \(r_1\) and \(r_2\) can be written as \((x - r_1)(x - r_2) = 0\). For the roots 2 and 5, the equation is \((x - 2)(x - 5) = 0\).

Step 3: Expand the Factored Equation

Multiply the two binomials \((x - 2)(x - 5)\): \((x - 2)(x - 5) = x \cdot x - 5x - 2x + 10 = x^2 - 7x + 10\).

Step 4: Write the Final Quadratic Equation

The expanded equation \(x^2 - 7x + 10 = 0\) is the quadratic equation with integer coefficients that has the roots 2 and 5.

Key Concepts

Roots of a Quadratic Equation

In the realm of quadratic equations, understanding what roots are is crucial. A quadratic equation typically appears in the form, \(ax^2 + bx + c = 0\). The solutions to this equation, known as roots, are the values of \(x\) that make the equation true. They are called roots because when substituted back into the equation, they result in zero. Each quadratic equation can have up to two real roots. These roots can be found using different methods such as factoring, using the quadratic formula, or completing the square. When given roots, they guide you in constructing a path back to the original equation. For instance, if you know the roots are \(2\) and \(5\), these values guide the formation of the quadratic equation. The relationship between the coefficients of the equation and its roots can also be explored using Vieta's formulas, which state that the sum of the roots is equal to \(-b/a\) and the product of the roots is \(c/a\). Understanding this helps when checking your work or when given non-integer roots.

Factored Form

The factored form of a quadratic equation simplifies finding the roots of the equation. When given the roots, you can express a quadratic equation as

• \((x - r_1)(x - r_2) = 0\),

where \(r_1\) and \(r_2\) are the roots of the equation. By constructing the equation in this manner, each binomial represents a linear equation that can be solved to find the roots. The factored form is particularly useful because it clearly shows the roots, making it straightforward to backtrack from the roots to the general quadratic form. For the roots \(2\) and \(5\), the equation crafted from them is \((x - 2)(x - 5) = 0\). This straightforward format tells you exactly where the solutions of the equation are. It's a compact way to represent the knowledge about the equation's solutions within the equation itself.

Expanding Binomials

Expanding binomials involves transforming a product of binomials into a polynomial. This process is essential when converting the factored form of a quadratic equation back into its standard form. To expand binomials, each term in the first binomial is multiplied by each term in the second binomial. To illustrate:

• Start with \((x - 2)(x - 5)\).
• Multiply \(x\) by \(x\), resulting in \(x^2\).
• Next, multiply \(x\) by \(-5\) to get \(-5x\).
• Then, \(-2\) by \(x\) gives you \(-2x\).
• Finally, \(-2\) by \(-5\) gives \(10\).

Combine these results to get \(x^2 - 5x - 2x + 10\). Simplifying by combining like terms, you find \(x^2 - 7x + 10\). This is the expanded form of the quadratic equation with integer coefficients that corresponds to the original roots of 2 and 5. Expanding binomials is a crucial step in bridging the factored form with the final expanded format.