Problem 11 - Precalculus Student Solutions Manual 5th

Question

Use Choices \(A-D\) to answer each question. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x-1)(x-7)=0\) Only one of the equations does not require Step 1 of the method for completing the square described in this section. Which one is it? Solve it.

Step-by-Step Solution

Answer

The equation \( (3x-1)(x-7)=0 \) does not need completing the square; solutions are \( x = \frac{1}{3} \) and \( x = 7 \).

Step 1 - Identify each equation format

Examine each equation to check if it is already set up as a factored equation or in the necessary form for solving.

Step 2 - Analyzing option D

The equation \( (3x-1)(x-7)=0 \) is already factored. This means we can use the zero-product property to solve it without completing the square.

Step 3 - Solve the factored equation

Set each factor equal to zero: \(3x - 1 = 0 \) and \(x - 7 = 0 \). Solve these equations to find the values of \(x\).

Step 4 - Solve for the first factor

For \(3x - 1 = 0\), add 1 to both sides to get \(3x = 1\). Then divide both sides by 3 to get \(x = \frac{1}{3}\).

Step 5 - Solve for the second factor

For \(x - 7 = 0\), add 7 to both sides to get \(x = 7\).

Step 6 - Combine the solutions

The solutions to the equation \( (3x-1)(x-7)=0 \) are \(x = \frac{1}{3}\) and \( x = 7 \).

Key Concepts

zero-product property

The zero-product property is a fundamental principle used to solve equations that are in factored form. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if \(a \times b = 0\), then \(a = 0\) or \(b = 0\).

For example, consider the equation \((3x-1)(x-7)=0\). Since the product is zero, either \(3x-1 = 0\) or \(x-7 = 0\).

When we solve these simpler equations individually, we find the solutions to the original equation. This property is extremely useful for solving quadratic equations when they are already factored, as it simplifies the process significantly.

Steps:

• Set each factor to zero: \( 3x - 1 = 0\) and \(x - 7 = 0\)
• Solve these equations for \(x\)

This will yield the values of \(x\) that satisfy the original equation.

solving quadratic equations

Quadratic equations often come in the form of \(ax^2 + bx + c = 0\). These equations can be solved through various methods:

• Factoring
• Completing the square
• Using the quadratic formula

One of the simplest methods is factoring, but this requires the equation to be factorable. In cases where the quadratic equation can be factored easily, it can directly lead to the solutions without additional steps.

For example, the equation \(3x^2 - 17x - 6 = 0 \) can often be factored into simpler binomials like \( (3x-1)(x-7)=0 \). Once factored, the zero-product property can be applied to directly solve for \(x\).

When it's difficult to factor, methods like completing the square or the quadratic formula are better suited. Completing the square involves transforming the equation into a perfect square trinomial, while the quadratic formula uses the coefficients \(a\), \(b\), and \(c\) to find the roots.

The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula is universally applicable to any quadratic equation.

factored form

Factored form of a quadratic equation looks like \((ax + b)(cx + d) = 0\). It is quite powerful because it allows you to easily apply the zero-product property.

Converting a quadratic equation to its factored form involves finding two binomials whose product is the original quadratic expression. For instance, \((3x-1)(x-7)=0\) is the factored form of a quadratic equation.

Steps to factor a quadratic equation:

• Look for common factors in all terms
• Write the equation as a product of two binomials
• Ensure the product of the binomials gives the original quadratic equation

For some quadratic equations, factoring them manually may be complex or not possible. In those cases, other methods to solve quadratic equations should be used.

The factored form, however, provides the most straightforward path to solutions when applicable, using the zero-product property. This form not only simplifies solving but also helps in understanding the roots' behavior and their geometric representation on a graph.