Problem 9 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ 3 x^{2}-5 x-2=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{3}\) and \(x = 2\).

Step 1: Identify the quadratic equation

The given quadratic equation is in the standard form: \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = -5\), and \(c = -2\).

Step 2: Multiply 'a' and 'c'

Calculate the product of \(a\) and \(c\), which are the coefficients of \(x^2\) and the constant term, respectively. Here, this is \(3 \times (-2) = -6\). This product will be used to factor the equation.

Step 3: Find two numbers that multiply to '-6' and add to '-5'

We are looking for two numbers whose product is \(-6\) and whose sum is \(-5\). These numbers are \(-6\) and \(1\), because \(-6 \times 1 = -6\) and \(-6 + 1 = -5\).

Step 4: Re-write the middle term using the two numbers

Re-write the equation by splitting the middle term, \(-5x\), into \(-6x + 1x\): \(3x^2 - 6x + x - 2 = 0\).

Step 5: Group and factor by grouping

Group the terms into pairs and factor each pair: \( (3x^2 - 6x) + (x - 2) = 0\). In the first group, factor out \(3x\): \(3x(x - 2)\), and in the second group, factor out \(1\): \(1(x - 2)\).

Step 6: Factor the common term 'x-2'

Since both groups have a common factor \((x - 2)\), factor \((x - 2)\) out: \((3x + 1)(x - 2) = 0\).

Step 7: Solve each factor for 'x'

Set each factor equal to zero and solve: - \(3x + 1 = 0\) gives \(x = -\frac{1}{3}\).- \(x - 2 = 0\) gives \(x = 2\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). It is characterized by having a degree of two, meaning that the highest power of the variable \(x\) is squared. In our exercise, the quadratic equation is \(3x^2 - 5x - 2 = 0\), where \(a = 3\), \(b = -5\), and \(c = -2\). Quadratic equations often have two solutions, known as roots, which can be real or complex numbers. Solving quadratic equations is a fundamental skill in algebra, as they appear in various mathematical contexts and practical applications.

Understanding how to manipulate and solve these can provide insights into more complex mathematical concepts.

Factoring by Grouping

Factoring by grouping is a method used to solve quadratic equations when direct factoring is not straightforward. This method involves rewriting a quadratic equation so that it can be grouped into pairs, which then can be factored further.

In our problem, we first find two numbers that multiply to the product of \(a\) and \(c\) (which is \(-6\)), and also add up to \(b\) (which is \(-5\)). Here, the numbers \(-6\) and \(1\) fit the criteria. The equation is rewritten by splitting the middle term:

• The original equation: \(3x^2 - 5x - 2 = 0\)
• Re-written as: \(3x^2 - 6x + 1x - 2 = 0\)

By grouping \((3x^2 - 6x)\) and \((1x - 2)\), we can factor them separately, allowing us to proceed with further factoring.

Solving Equations

Solving equations involves finding the values of \(x\) that make the equation true. In this context, once an equation is expressed in a factored form, solving it becomes more straightforward. After factoring the equation \((3x + 1)(x - 2) = 0\), each factor is set to zero, resulting in two linear equations to solve:

• For \(3x + 1 = 0\), solving it gives: \(x = -\frac{1}{3}\).
• For \(x - 2 = 0\), solving it gives: \(x = 2\).

These solutions \(x = -\frac{1}{3}\) and \(x = 2\) are the roots of the quadratic equation. This approach uses the Zero Product Property, which states that if a product equals zero, then at least one of the factors must be zero.

Roots of a Polynomial

The roots of a polynomial are the values for which the polynomial equals zero. In the case of a quadratic equation, there are typically two roots. These roots can be real or complex, depending on the coefficients \(a\), \(b\), and \(c\).

In our exercise, the roots were calculated after factoring by grouping and solving the resulting linear equations. The roots \(x = -\frac{1}{3}\) and \(x = 2\) indicate where the graph of the quadratic function \(y = 3x^2 - 5x - 2\) will intersect the x-axis. Identifying the roots also provides insight into the symmetry and vertex of the parabola represented by the quadratic. Understanding the roots of a polynomial allows us to explore further properties and behaviors of quadratic functions.