Problem 21 - Basic Technical Mathematics with Calculus

Question

$$\text { Solve the given quadratic equations by factoring.}$$ $$3 x^{2}-13 x+4=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3}\) and \(x = 4\).

Step 1: Identify and Verify Quadratic Form

First, we ensure that the given quadratic equation, \(3x^2 - 13x + 4 = 0\), is in standard form \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = -13\), and \(c = 4\).

Step 2: Factor the Quadratic Equation

Find two numbers that multiply to \(a \times c = 3 \times 4 = 12\) and add up to \(b = -13\). These numbers are \(-12\) and \(-1\), as \((-12) \times (-1) = 12\) and \((-12) + (-1) = -13\).

Step 3: Re-write Middle Term

Re-write the equation by splitting the middle term using the numbers found: \[ 3x^2 - 12x - x + 4 = 0 \]

Step 4: Factor by Grouping

Group the terms in pairs and factor out the common factors in each group: \[ (3x^2 - 12x) + (-x + 4) = 0 \]Factor each group: \[3x(x - 4) - 1(x - 4) = 0 \]

Step 5: Factor the Common Binomial

Notice \((x - 4)\) is a common factor:\[(3x - 1)(x - 4) = 0 \]

Step 6: Solve for Roots

Set each factor equal to zero to solve for \(x\):1. \(3x - 1 = 0\) leads to \(x = \frac{1}{3}\).2. \(x - 4 = 0\) leads to \(x = 4\).

Key Concepts

Factoring Quadratics

Factoring quadratics involves breaking down a quadratic expression into the product of two simpler binomial expressions. For instance, in the quadratic equation \(3x^2 - 13x + 4 = 0\), factoring helps us rewrite the equation as \((3x - 1)(x - 4) = 0\). This makes it easier to find the values of \(x\) that satisfy the equation.

Here's how you can factor a quadratic equation step by step:

• First, ensure the equation is in the form \(ax^2 + bx + c = 0\). Identify the coefficients \(a\), \(b\), and \(c\).
• Calculate \(ac\) (the product of \(a\) and \(c\)).
• Find two numbers that multiply to \(ac\) and add up to \(b\).
• Use these numbers to split the middle term \(bx\) into two terms.
• Group the terms into pairs and factor each pair.
• Factor out the common binomial from the resulting expression.

This approach is a reliable method to break down complex quadratic expressions into simpler components, making it straightforward to find their roots.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential aspect of solving quadratics. It's represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This format is crucial because it lays the groundwork for various methods of solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.

For the equation \(3x^2 - 13x + 4 = 0\), you can see it's already in the standard form:

• \(a = 3\)
• \(b = -13\)
• \(c = 4\)

Being in standard form makes it easy to identify the necessary components for factoring or applying other solving techniques.

Roots of Quadratic Equations

The roots of a quadratic equation are the solutions to the equation, which satisfy the condition where the equation equates to zero. In the context of the factored form \((3x - 1)(x - 4) = 0\), solving for the roots involves setting each factor equal to zero.

• For \(3x - 1 = 0\), solve for \(x\):
  \[3x = 1\]
  \[x = \frac{1}{3}\]
• For \(x - 4 = 0\), solve for \(x\):
  \[x = 4\]

Thus, the roots of the quadratic equation \(3x^2 - 13x + 4 = 0\) are \(x = \frac{1}{3}\) and \(x = 4\). These roots are where the graph of the quadratic would cross the x-axis, commonly referred to as the x-intercepts.