Problem 12 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(x^{2}-\frac{1}{4}=0\)

Step-by-Step Solution

Answer

Sum of roots: 0; Product of roots: -1/4.

Step 1: Identify Equation Format

The given equation is in the standard quadratic form of \(ax^2 + bx + c = 0\). We start with \(x^2 - \frac{1}{4} = 0\), where \(a = 1\), \(b = 0\), and \(c = -\frac{1}{4}\).

Step 2: Apply Sum of Roots Formula

For any quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(\alpha + \beta\) can be calculated using the formula \(-\frac{b}{a}\). Thus, the sum of the roots is \(-\frac{0}{1} = 0\).

Step 3: Apply Product of Roots Formula

For the same quadratic equation, the product of the roots \(\alpha \beta\) is given by \(\frac{c}{a}\). Therefore, the product of the roots is \(\frac{-\frac{1}{4}}{1} = -\frac{1}{4}\).

Step 4: Verify with Vieta's Formulas

Using Vieta's formulas, which state that for a quadratic equation \(x^2 + px + q = 0\), the sum of the roots \(\alpha + \beta = -p\) and the product \(\alpha \beta = q\), verify that the calculated sum \(0\) and product \(-\frac{1}{4}\) are consistent with \(b = 0\) and \(c = -\frac{1}{4}\). This confirms the calculations.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in algebra and are widely used in various fields of mathematics and science. A quadratic equation is any polynomial equation where the highest power of the variable is 2. They generally take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).

Each quadratic equation can have two solutions, known as roots. These roots are the values of \(x\) that make the equation equal to zero. The nature of the roots (whether they are real or complex) depends on the discriminant, \(b^2 - 4ac\).

When working with quadratic equations, it's helpful to understand their graphical representation. The graph of a quadratic equation is a parabola, which opens upward if \(a > 0\) and downward if \(a < 0\). Parabolas have key features like vertices and axes of symmetry. Being familiar with these will aid in grasping more complex aspects of algebra.

Exploring Vieta's Formulas

Vieta's formulas offer a simple yet powerful way to understand the relationship between the coefficients of a polynomial and its roots, without actually finding the roots. For quadratic equations, they can be particularly useful. If we have a quadratic equation of the form \(x^2 + px + q = 0\), Vieta's formulas tell us two things:

• The sum of the roots \((\alpha + \beta)\) is \(-p\).
• The product of the roots \((\alpha \beta)\) is \(q\).

Vieta's formulas provide an efficient way to find the sum and product of the roots directly from the equation’s coefficients, without solving the equation explicitly.

In the context of the given equation \(x^2 - \frac{1}{4} = 0\), by rearranging it to match the typical form \(x^2 + 0x - \frac{1}{4} = 0\), Vieta's formulas verify that the sum of the roots is zero and their product is \(-\frac{1}{4}\). This can be extremely handy in mathematical proofs and when dealing with higher order polynomials where direct computation of roots is cumbersome.

Delving into the Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation. It provides a way to find the roots of any quadratic equation \(ax^2 + bx + c = 0\). The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is derived from completing the square of the quadratic equation and it allows for the computation of complex roots when the discriminant \((b^2 - 4ac)\) is negative.

Using the quadratic formula involves plugging in the values of \(a\), \(b\), and \(c\) into the formula, which then gives the two roots of the equation. It provides a reliable method for solving quadratics and can verify results derived from other methods like factorization or completing the square.

In our given example \(x^2 - \frac{1}{4} = 0\), by substituting \(a = 1\), \(b = 0\), and \(c = -\frac{1}{4}\), we can find the roots directly using the quadratic formula, further confirming the sum and product of roots determined earlier via Vieta's methods.