Problem 46 - Algebra 2

Question

Write a quadratic equation with the given graph or roots. \(-\frac{3}{2},-\frac{4}{5}\)

Step-by-Step Solution

Answer

The quadratic equation is \(10x^2 + 23x + 12 = 0\).

Step 1: Understand the Problem

The problem provides us with two roots of a quadratic equation: \(-\frac{3}{2}\) and \(-\frac{4}{5}\). We need to write a quadratic equation that has these two roots.

Step 2: Use the Factored Form of the Quadratic Equation

A quadratic equation with roots \(r_1\) and \(r_2\) can be written as \((x - r_1)(x - r_2) = 0\). Here, \(r_1 = -\frac{3}{2}\) and \(r_2 = -\frac{4}{5}\). Thus, the equation is:\[(x - (-\frac{3}{2}))(x - (-\frac{4}{5})) = 0\]Simplifying, we have:\[(x + \frac{3}{2})(x + \frac{4}{5}) = 0\]

Step 3: Expand the Quadratic Equation

Now, we expand \((x + \frac{3}{2})(x + \frac{4}{5})\):\[x^2 + \left(x \cdot \frac{4}{5} + x \cdot \frac{3}{2}\right) + \frac{3}{2} \cdot \frac{4}{5}\]Combining the middle terms: \(x \cdot \frac{4}{5} + x \cdot \frac{3}{2}\) becomes:\[\frac{8}{10}x + \frac{15}{10}x = \frac{23}{10}x\]The constant term \(\frac{3}{2} \cdot \frac{4}{5}\) simplifies to:\[\frac{12}{10} = \frac{6}{5}\]Thus, the expanded quadratic equation is:\[x^2 + \frac{23}{10}x + \frac{6}{5} = 0\]

Step 4: Write in Standard Form

Typically, a quadratic equation is written in the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are integers. To eliminate the fractions, multiply the entire equation by 10:\[10x^2 + 23x + 12 = 0\]This is the final quadratic equation in standard form.

Key Concepts

Factored Form

Understanding the factored form of a quadratic equation is crucial when you have the roots and need to create the equation itself. If you have roots, say \( r_1 \) and \( r_2 \), the factored form follows this structure:

• \( (x - r_1)(x - r_2) = 0 \).

This configuration of factors allows you to directly insert the given roots into the equation. For instance, in the exercise provided, the roots are \( -\frac{3}{2} \) and \( -\frac{4}{5} \). Substituting these into the factored form leads to:

• \( (x + \frac{3}{2})(x + \frac{4}{5}) = 0\).

Inserting the negatives of the roots might initially seem puzzling, but remember, each binomial within the factored form is derived by solving \( x = r \). Therefore, subtracting the root brings it to the right-hand side of the equation. This factored form sets a groundwork for further operations, such as expanding.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \( x \) that satisfy the equation when it is set to zero. In simple terms, these are the x-values where the parabola intersects the x-axis. Finding the roots is a fundamental step in solving quadratics.

• It indicates the solutions of the equation.
• It helps in formulating the factored form of the equation.

Given the roots -\frac{3}{2} and -\frac{4}{5}, you immediately know that if the equation is zero, \( x \) must be:

• \-\frac{3}{2}\ or \-\frac{4}{5}\.

It is these points on the x-axis that define the characteristics of the quadratic graph. Including these roots in your quadratic formulation helps rewrite the equation into its factored form, making further operations such as expansion straightforward.

Expanding Expressions

Expanding expressions involves multiplying out the terms in a product to remove parentheses and write the expression as a sum or difference of individual terms. This is an essential skill in algebra that is particularly useful for writing quadratics in standard form.
Let's take our previously formed factored expression, \( (x + \frac{3}{2})(x + \frac{4}{5})\). To expand, you apply the distributive property:

• Multiply each pair of terms from the two binomials.
• Add together the resulting terms.

The result is a quadratic with each component expanded:

• \x^2\ (from multiplying \( x \) terms),
• \+ \frac{23}{10}x\ (combining middle terms),
• \+ \frac{6}{5}\ (constant product).

Expanding expressions is useful not just for converting from factored to expanded form but also in preparing the quadratic for presentation in standard form, i.e., \( ax^2 + bx + c = 0\). By having a clear understanding of these steps, solving quadratic equations becomes an achievable task.