Problem 45 - Algebra 2

Question

Write a quadratic equation with the given graph or roots. \(-\frac{2}{3}, \frac{3}{4}\)

Step-by-Step Solution

Answer

The quadratic equation with roots \(-\frac{2}{3}\) and \(\frac{3}{4}\) is \(12x^2 - x - 6 = 0\).

Step 1: Identify Roots

The given roots of the quadratic equation are \(-\frac{2}{3}\) and \(\frac{3}{4}\). A quadratic equation with these roots will have the form \(a(x - \text{{root 1}})(x - \text{{root 2}}) = 0\).

Step 2: Set Up Equation with Roots

Using the roots \(-\frac{2}{3}\) and \(\frac{3}{4}\), the equation becomes \( a (x + \frac{2}{3})(x - \frac{3}{4}) = 0\). We choose \(a = 12\) to eliminate fractions, which simplifies the arithmetic.

Step 3: Expand and Simplify

First, find the product \((x + \frac{2}{3})(x - \frac{3}{4})\):\[x^2 - \frac{3}{4}x + \frac{2}{3}x - \frac{1}{2} = x^2 - \frac{3}{4}x + \frac{2}{3}x - \frac{1}{2}\]Combine like terms (convert fractions to a common denominator):\[x^2 - \frac{9}{12}x + \frac{8}{12}x - \frac{1}{2} = x^2 - \frac{1}{12}x - \frac{1}{2}\]Simplify further by multiplying by \(12\):\[12(x^2 - \frac{1}{12}x - \frac{1}{2}) = 12x^2 - x - 6 = 0\]

Step 4: Write Final Quadratic Equation

The quadratic equation with roots \(-\frac{2}{3}\) and \(\frac{3}{4}\), in standard form, is:\[12x^2 - x - 6 = 0\]

Key Concepts

Roots of a Quadratic

Understanding the roots of a quadratic equation is a key part of solving or forming these equations. The roots, also known as zeros or solutions, are the values of \(x\) that make the quadratic equation equal to zero. For the equation \(ax^2 + bx + c = 0\), finding the roots means determining the values of \(x\) where the equation holds true.

If you have the roots provided in the problem, like \(-\frac{2}{3}\) and \(\frac{3}{4}\), these tell us where the graph of the quadratic crosses the x-axis. Knowing just the roots allows you to write the equation in factored form, such as \(a(x - \text{root 1})(x - \text{root 2}) = 0\). When given specific roots, this process can help in constructing the quadratic equation. Remember, roots also give insights into the symmetry and shape of the quadratic equation's graph.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to express the equation as a product of its linear factors. This method is particularly handy when we're given roots and need to find the corresponding quadratic equation.

For example, with roots \(-\frac{2}{3}\) and \(\frac{3}{4}\), you set up the equation \(a(x + \frac{2}{3})(x - \frac{3}{4}) = 0\). By factoring this way, you are re-expressing the quadratic equation based on its roots. Simplifying involves:

• Combining like terms if necessary.
• Ensuring any arithmetic or algebraic manipulation keeps the essence of the equation unchanged.
• Multiplying by a common denominator, like \(12\), to avoid fractions and simplify the equation.
  
  Factoring helps find the simplest form of the quadratic, making it easier to analyze or solve.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. This form is pivotal for identifying specific properties of the quadratic equation, such as its direction of opening, vertex, and axis of symmetry.

From the factored form \(12(x + \frac{2}{3})(x - \frac{3}{4}) = 0\), you can expand to turn it into its standard form. Let's see how:
Start by multiplying the factors and then adjusting coefficients by necessary multiplication for simplification, such as multiplying by \(12\) to eliminate fractions. This results in:

• \(12x^2 - x - 6 = 0\)

This process translates the equation into a familiar format that can be easily used in further algebraic procedures or graph plotting. Standard form is crucial for solving via methods like the quadratic formula as well.