Problem 12 - 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 is set up so that the values of \(a, b,\) and \(c\) can be determined immediately. Which one is it? Solve it.

Step-by-Step Solution

Answer

Equation A is suitable. The solutions are \(x = 6\) and \(x = -\frac{1}{3}\)

Step 1: Identify the standard quadratic equation

A standard quadratic equation is in the form of \(ax^2 + bx + c = 0\). We need to identify which one of the given equations matches this form directly.

Step 2: Analyze each equation

Let's check each option: A. \(3x^2 - 17x - 6 = 0\) matches the standard form.B. \((2x+5)^2 = 7\) does not match the standard form.C. \(x^2 + x = 12\) does not match the standard form.D. \((3x-1)(x-7) = 0\) needs to be expanded to match the standard form.

Step 3: Select the equation that fits the standard form

Based on the analysis, option A fits the standard quadratic form directly: \(3x^2 - 17x - 6 = 0\). So, identifying the coefficients: \(a = 3\), \(b = -17\), \(c = -6\).

Step 4: Solve the quadratic equation by using the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the values of \(a\), \(b\), and \(c\): \[ a = 3 \] \[ b = -17 \] \[ c = -6 \]. So: \ x = \frac{17 \pm \sqrt{(-17)^2-4(3)(-6)}}{2(3)} = \frac{17 \pm \sqrt{289 + 72}}{6} = \frac{17 \pm \sqrt{361}}{6} = \frac{17 \pm 19}{6} .= \frac{17+19}{6} \or \frac{17-19}{6}x = 6\ or\ x = -\frac{1}{3}\ = 6 = -\frac{1}{3}

Key Concepts

Standard Quadratic Form

Before solving any quadratic equation, it's important to understand its standard form. A quadratic equation is in the standard form if it looks like this:
\(ax^2 + bx + c = 0\)
Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The main components are:

• \(ax^2\) is the quadratic term.
• \(bx\) is the linear term.
• \(c\) is the constant term.

To identify an equation in standard form, it must match exactly \(ax^2 + bx + c = 0\). For example, the equation \(3x^2 - 17x - 6 = 0\) fits the standard form directly. This makes it easy to identify the values of \(a\), \(b\), and \(c\) which are 3, -17, and -6 respectively. Recognizing this form is the first step towards solving the equation efficiently.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It's derived from the standard form and helps find the values of \(x\). The formula is:
\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
This formula provides solutions for any quadratic equation, as long as the equation is in the standard form \(ax^2 + bx + c = 0\). Let's break down the components:

• \( -b \) is the negation of the linear term's coefficient.
• \( \sqrt{b^2 - 4ac} \) is the discriminant, determining the nature of the roots.
• \( 2a \) is the denominator that normalizes the equation.

By substituting \(a=3\), \(b=-17\), and \(c=-6\) into the quadratic formula, we get:
\[x = \frac{{17 \pm \sqrt{{(-17)^2 - 4(3)(-6)}}}}{2(3)}\]
Simplifying further gives us:
\[x = \frac{{17 \pm \sqrt{361}}}{6}\]
This calculation results in two solutions for \(x\):
\[ x = \frac{17 + 19}{6} = 6, \quad x = \frac{17 - 19}{6} = -\frac{1}{3} \]
Understanding and correctly applying the quadratic formula is vital for successfully solving quadratic equations.

Solving Quadratic Equations

Once we understand the standard form and the quadratic formula, solving quadratic equations becomes straightforward. Here’s a step-by-step process to solve any quadratic equation:

• Step 1: Write the Equation in Standard Form: Ensure the equation is in the form \(ax^2 + bx + c = 0\). If not, manipulate it to this form. For example, for the equation \((3x-1)(x-7) = 0\), you need to expand it.


• Step 2: Identify Coefficients: Determine the values of \(a\), \(b\), and \(c\). For instance, in \(3x^2 - 17x - 6 = 0\), we have \(a = 3\), \(b = -17\), and \(c = -6\).


• Step 3: Apply the Quadratic Formula: Use the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). Substitute the coefficients into the equation.


• Step 4: Simplify to Find Solutions: Solve the equation inside the square root, then simplify the expression to find the precise values for \(x\).

Example: Solve \(3x^2 - 17x - 6 = 0\) using the quadratic formula.
Step 4 detailed solution from the exercise shows the correct final values of \( x = 6 \) and \( x = -\frac{1}{3} \). Practicing these steps with various problems will improve your ability to solve quadratic equations proficiently. Remember to stay consistent with the steps and double-check your arithmetic to avoid mistakes.