Problem 24 - Precalculus: Functions and Graphs

Question

Use the quadratic formula to factor the expressions. \(15 x^{2}+34 x-16\)

Step-by-Step Solution

Answer

The expression factors as \((5x - 2)(3x + 8)\).

Step 1: Identify the Coefficients

The quadratic expression is given in the form \(ax^2 + bx + c\). Identify \(a = 15\), \(b = 34\), and \(c = -16\).

Step 2: Write the Quadratic Formula

Recall the quadratic formula for finding the roots of \(ax^2 + bx + c = 0\) which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Compute the discriminant \(b^2 - 4ac\). Here it is \(34^2 - 4 \times 15 \times (-16)\). Thus, compute: 1156 + 960 = 2116.

Step 4: Take the Square Root of the Discriminant

Find the square root of 2116. \(\sqrt{2116} = 46\).

Step 5: Solve for Roots

Substitute the values into the quadratic formula:\[x = \frac{-34 \pm 46}{30}\] Calculate the two possible roots:1. \(x = \frac{-34 + 46}{30} = \frac{12}{30} = \frac{2}{5}\)2. \(x = \frac{-34 - 46}{30} = \frac{-80}{30} = \frac{-8}{3}\).

Step 6: Write the Factored Form Using Roots

Transform the expression using the roots to its factored form. The factored expression is:\[15(x - \frac{2}{5})(x + \frac{8}{3})\]Ensure the expression is in integer form by multiplying through by 5 and 3:\[(5x - 2)(3x + 8)\].

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental skill in algebra that allows you to write a quadratic equation in a simpler, more useful form. A quadratic equation typically appears in the form \(ax^2 + bx + c\). Here, we're looking to express it in factored form: \(a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots of the equation.
Factoring helps in simplifying equations, making them easier to solve or graph. The process often begins by identifying the coefficients \(a\), \(b\), and \(c\) of the quadratic expression, as was done with the expression \(15x^2 + 34x - 16\). Once you have the roots, these can be used to write the equation in its factored form. For instance, roots \( \frac{2}{5} \) and \( -\frac{8}{3} \) yield the factors \((5x - 2)\) and \((3x + 8)\).
Factoring is beneficial because it is useful for solving equations and helps with understanding the behavior of quadratic functions. Knowing how to factor quadratics effectively can enhance your skill in analyzing mathematical problems.

Quadratic Roots

Quadratic roots are the solutions to the quadratic equation set equal to zero. When we solve \(ax^2 + bx + c = 0\) using the quadratic formula, we are essentially finding the roots. These roots can be real or complex numbers, depending on the value of the discriminant.
The process of finding roots involves:

• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  
• Substituting in the values of \(a\), \(b\), and \(c\). For example, for \(15x^2 + 34x - 16\), calculating \(\frac{-34 \pm 46}{30}\) resulted in the roots \(\frac{2}{5}\) and \(-\frac{8}{3}\).

Understanding the roots of quadratic equations is crucial as it gives insight into the value where the function crosses the \(x\)-axis (i.e., the solutions of the equation). Knowing the roots gives a clearer picture of the function's behavior and assists in graphing.

Discriminant Calculation

The discriminant is a key component of the quadratic formula and plays a critical role in determining the nature of roots a quadratic equation has. For any quadratic equation given by \(ax^2 + bx + c = 0\), the discriminant is calculated using the formula \(b^2 - 4ac\).
The discriminator provides valuable information:

• If the discriminant is positive, there are two distinct real roots. For the equation \(15x^2 + 34x - 16\), the discriminant calculated as \(2116\) is positive, thus producing two real roots.
• If the discriminant is zero, there is exactly one real root (or a repeated real root).
• If the discriminant is negative, there are two complex roots, meaning the roots are not real numbers.

Calculating the discriminant helps anticipate the type and number of solutions the quadratic equation possesses before actually computing them, thus making it an essential initial step in solving quadratic equations. Knowing whether the roots are real or complex lets you decide how best to tackle the equation.