Constructing a Quadratic Equation with Given Roots: A Comprehensive Guide

Understanding how to construct a quadratic equation with specific roots is a fundamental concept in algebra. This guide will walk you through the process step-by-step, explaining the underlying principles and providing a real example.

The Theory of Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of such an equation is ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. One of the key features of a quadratic equation is that it has two roots, which are the values of x that satisfy the equation.

Constructing a Quadratic Equation from Given Roots

Given a set of roots, we can construct a quadratic equation by expressing it in its factored form and then expanding it. The factored form of a quadratic equation with roots r_1 and r_2 is:

A(x - r_1)(x - r_2) 0

Example with Specific Roots

Let's construct a quadratic equation with the roots 2/3 and -1/2 using the given formula.

Step 1: Express the Equation in Factored Form

Given the roots, the equation can be expressed as:

A(x - 2/3)(x 1/2) 0

Step 2: Choose a Simplifying Constant

For simplicity, we can choose A 1, although any non-zero constant would work. Thus, our equation becomes:

(x - 2/3)(x 1/2) 0

Step 3: Expand the Expression

Expand the brackets to get the standard form of the quadratic equation:

x(x 1/2) - 2/3(x 1/2) 0

Step 3.1: Distributive Property

Apply the distributive property:

x^2 1/2x - 2/3x - 2/3(1/2) 0

Step 3.2: Combine Like Terms

Combine the like terms, first converting all terms to a common denominator:

x^2 (1/2 - 2/3)x - 1/3 0

Convert 1/2 to sixths:

1/2 3/6

Combine the fractions:

x^2 (3/6 - 4/6)x - 1/3 0

Simplify the expression:

x^2 - 1/6x - 1/3 0

Step 4: Clear the Fractions

To eliminate the fractions, multiply the entire equation by 6:

6(x^2 - 1/6x - 1/3) 0

Expand and simplify:

6x^2 - x - 2 0

Thus, the quadratic equation with roots 2/3 and -1/2 is:

6x^2 - x - 2 0

General Case Consideration

We can generalize the process for any pair of roots α and β. The quadratic equation with roots α and β can be written as:

(x - α)(x - β) 0

Expanding this, we get:

x^2 - (α β)x αβ 0

This form is known as the standard form of a quadratic equation given its roots.

Verification Using the Quadratic Formula

To verify, let's apply the quadratic formula to our quadratic equation:

The general quadratic formula is:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

For our quadratic equation 6x^2 - x - 2 0, we have:

a 6, b -1, c -2

Substitute these values into the quadratic formula:

x [1 ± sqrt((-1)^2 - 4(6)(-2))] / (2(6))

Simplify the expression:

x [1 ± sqrt(1 48)] / 12

x [1 ± sqrt(49)] / 12

x [1 ± 7] / 12

This gives us two solutions:

x (1 7) / 12 8 / 12 2/3

x (1 - 7) / 12 -6 / 12 -1/2

These are the roots we started with, confirming the solution is correct.

Understanding the construction of quadratic equations from given roots is essential in algebra and can be applied in various mathematical and scientific contexts.