Problem 68 - Algebra 2

Question

Write a quadratic equation with the given roots. Write the equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are integers. \(6,4\)

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 - 10x + 24 = 0 \).

Step 1: Understand the Root-Vieta's Formula

A quadratic equation with roots \( p \) and \( q \) can be expressed using Vieta's formulas as \( x^2 - (p+q)x + pq = 0 \). Here, \( p = 6 \) and \( q = 4 \).

Step 2: Calculate the Sum of the Roots

Use the sum of the roots formula: \( p+q = 6 + 4 = 10 \). This value will be used as the coefficient of the \( x \) term in the equation.

Step 3: Calculate the Product of the Roots

Use the product of the roots formula: \( pq = 6 \times 4 = 24 \). This value will be the constant term (\( c \)) in the equation.

Step 4: Construct the Quadratic Equation

Substitute the sum and product of the roots into the quadratic formula from Step 1: \( x^2 - 10x + 24 = 0 \). This is the desired quadratic equation with integer coefficients.

Key Concepts

Vieta's Formulas

Vieta's formulas are powerful tools for understanding the relationships between the roots and coefficients of a polynomial. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us:

• The sum of the roots \((p + q)\) is given by \(-b/a\).
• The product of the roots \((pq)\) is equal to \(c/a\).

These relationships emerge directly from the properties of the polynomial and offer a way to construct a quadratic equation if the roots are known.
In our exercise, the roots were given as 6 and 4. By using Vieta's formulas, - the sum of the roots is 10, which corresponds to the coefficient of the \(x\) term, and - the product of the roots is 24, which is the constant term. These connections allow us to write the equation \(x^2 - 10x + 24 = 0\), effectively tying the roots to the coefficients.

Roots of a Quadratic

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). There are usually two such roots for a quadratic equation, which can be real or complex numbers depending on the discriminant. The roots can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In the context of our exercise, the roots 6 and 4 were provided, indicating these are the values of \(x\) that would solve the equation \(x^2 - 10x + 24 = 0\).
Understanding the roots lays the foundation for interpreting what the quadratic equation means. When the graph of a quadratic, a parabola, crosses the x-axis, the x-coordinates of these points are the roots of the equation. In this particular problem, the parabola intersects the x-axis at \(x = 6\) and \(x = 4\). This visible intersection makes it easier to understand how the roots relate back to the quadratic equation.

Integer Coefficients

When dealing with quadratic equations, the coefficients \(a\), \(b\), and \(c\) are essential as they define the specific shape and position of the parabola related to the equation. Oftentimes, it's crucial or desirable for these coefficients to be integers. This requirement simplifies calculations and is particularly important in scenarios like our exercise, where the resulting equation must maintain integer coefficients.
In our problem, we constructed the quadratic equation from given roots 6 and 4 using Vieta’s formulas. As a result, we ensured that all coefficients (1 for \(a\), -10 for \(b\), and 24 for \(c\)) are integers.

• The leading coefficient \(a = 1\) simplifies the equation, keeping all operations straightforward.
• The sum and product of the roots directly contribute to coefficients \(b\) and \(c\) without introducing fractional values.

This strategy not only makes the equation more elegant but also its invariants more apparent, allowing us to see the link between roots and coefficients clearly. Keeping coefficients as integers is especially helpful in a learning context, fostering a clearer understanding of the roots and equation relationship.