Problem 77 - Algebra A Combined Function

Question

Write a quadratic equation that has two solutions, 6 and -1 . Leave the polynomial in the equation in factored form.

Step-by-Step Solution

Answer

The equation is (x - 6)(x + 1) = 0 .

Step 1: Understand Roots of the Equation

The roots of the quadratic equation are given as 6 and -1. Knowing that the solutions are the values for which the equation equals zero ( x = 6 and x = -1 ), we will construct the equation from these solutions.

Step 2: Express Roots Using Factors

The factored form of a quadratic equation with roots 6 and -1 can be expressed as (x - 6) and (x + 1) . These factors represent the points where the equation equals zero.

Step 3: Write the Equation in Factored Form

Write the quadratic equation using the factors from Step 2: (x - 6)(x + 1) = 0 . This is the quadratic equation in its factored form.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and are characterized by their polynomial form:

• General form: \( ax^2 + bx + c = 0 \)
• Degree: A quadratic equation has a degree of 2, which means the highest power of the variable \( x \) is 2.

These equations are called 'quadratic' because 'quadra' means square. The quadratic part refers to the squaring of the variable. Quadratic equations pop up in many areas of mathematics, physics, and engineering because they describe parabolic relationships.
They are essential when dealing with areas, projectile motions, and optimizing specific conditions in real-world problems. Studying their general and nature specific forms helps solve numerous equations in mathematical representations.
In this section, we are especially interested in expressing them in factored form so as to easily identify the roots, which are the solutions to these equations.

Roots of Equations

The roots of an equation, often referred to as solutions or zeros, are values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
These solutions are the values where the quadratic equation equals zero.

• For a quadratic equation, there can be a maximum of two roots.
• Root terms come into play with this equation form as these determine the direction and position where the parabola intersects the x-axis in its graphical representation.

In the context of our exercise, the roots given are 6 and -1. This means:

• \( x = 6 \)
• \( x = -1 \)

Each root represents an x-value at which the equation's corresponding y-value, \((f(x))\), becomes zero.
Understanding roots is crucial for factoring and solving quadratic equations because they directly determine the form of the equation.

Factored Form of Quadratics

The factored form of a quadratic equation is a powerful way to express such equations because it directly facilitates finding the roots.

• Factored form: \((x - p)(x - q) = 0\), where \(p\) and \(q\) are the roots of the equation.
• This reveals immediate x-values where the graph of the quadratic will cross the x-axis.

In our exercise, we learned that the roots are 6 and -1.
This allows us to write the factored form as \((x - 6)(x + 1) = 0\).

• This expression clearly tells us that replacing \(x\) with 6 or -1 will result in zero.
• Each factor goes to zero at its respective root, simplifying the process of solving the quadratic equation.

By writing a quadratic in its factored form, we gain direct insight into its solutions,
enabling easier analysis and understanding of the equation's behavior.