Problem 15 - Algebra 2

Question

Write a quadratic equation in standard form with the given roots. \(4,-5\)

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + x - 20 = 0\).

Step 1: Understand the Relationship between Roots and Equation

To write a quadratic equation with given roots, recall that if the roots are \(r_1\) and \(r_2\), the equation can be represented as \((x - r_1)(x - r_2) = 0\). For the roots 4 and -5, this becomes \((x - 4)(x + 5) = 0\).

Step 2: Expand the Equation

Now, expand the expression \((x - 4)(x + 5)\) to write the quadratic equation in expanded form. Start with: \((x - 4)(x + 5) = x(x + 5) - 4(x + 5)\) Use the distributive property to simplify: \(x^2 + 5x - 4x - 20\)

Step 3: Combine Like Terms

Combine the like terms in the equation \(x^2 + 5x - 4x - 20\) which simplifies to: \(x^2 + x - 20\).

Step 4: Write the Equation in Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). The expanded and simplified equation is already in this form: \(x^2 + x - 20 = 0\). Thus, the quadratic equation in standard form with roots 4 and -5 is \(x^2 + x - 20 = 0\).

Key Concepts

Roots of a Quadratic

In a quadratic equation, the roots are the solutions for the variable when the equation equals zero. Think of the roots as the points where the parabola, a U-shaped curve representing the equation, crosses the x-axis on a graph. Thus, if you know where the parabola intersects the x-axis, you can identify the roots.

For example, if we have a quadratic equation with roots 4 and -5, it means these two numbers make the equation zero when substituted for the variable. In simpler terms, the roots of 4 and -5 make the equation ((x - 4)(x + 5) = 0). By knowing this, we can "work backwards" to construct the entire equation from these roots.

Standard Form

The standard form of a quadratic equation is a universally recognized structure: (ax^2 + bx + c = 0). In this form, 'a', 'b', and 'c' are constants, and 'x' is the variable.

• The 'a' term represents the coefficient of the squared term (x^2).
• The 'b' term is the coefficient of the linear term (x).
• The 'c' term is the constant, or the number by itself.

A quadratic equation in standard form succinctly communicates the key parts of the equation, making it easier to analyze or graph. In our example with roots 4 and -5, we found that the standard form of the resulting equation is (x^2 + x - 20 = 0). This means the equation has been neatly organized to highlight its inherent structure and components.

Distributive Property

The distributive property is a fundamental mathematical law that allows you to simplify expressions by distributing one term across terms inside a bracket. It's key in expanding expressions like ((x - 4)(x + 5)).

Here's how it works in our example:

• First: Multiply the first term in (x - 4) by both terms in (x + 5) to get (x^2 + 5x).
• Then: Multiply the -4 by (x + 5) to get (-4x - 20).
• Combine these products: (x^2 + 5x - 4x - 20).

After using the distributive property, you simplify by combining like terms, which leads you to the neat equation (x^2 + x - 20), demonstrating an efficient method to transition from a product of binomials into a standard form quadratic equation.