Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Only one of the following equations is set up so that the values of \(a, b,\) and \(c\) can be determined immediately. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

Step-by-Step Solution

Answer

Equation A allows immediate identification: \(a = 3\), \(b = -17\), \(c = -6\).

Step 1: Analyze Equation A

Equation A is \(3x^2 - 17x - 6 = 0\). This is a quadratic equation in standard form \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = -17\), and \(c = -6\). While \(a, b,\) and \(c\) are easily identified, the values of \(x\) still need solving by factoring or using the quadratic formula.

Step 2: Analyze Equation B

Equation B is \((2x+5)^2 = 7\). This equation requires expansion or solving by taking square roots, making it less straightforward to immediately determine \(a, b,\) and \(c\).

Step 3: Analyze Equation C

Equation C is \(x^2 + x = 12\). To find \(a, b,\) and \(c\), we first need to rearrange this into the standard form: \(x^2 + x - 12 = 0\). Here, \(a = 1\), \(b = 1\), and \(c = -12\). While its coefficients can now be identified, it also needs solving.

Step 4: Analyze Equation D

Equation D is \((3x+1)(x-7) = 0\). This equation is already factored, so the solutions for \(x\) can be immediately determined from \(3x+1=0\) and \(x-7=0\), but it doesn't fit the form \(ax^2 + bx + c = 0\) to directly identify \(a, b,\) and \(c\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing the quadratic as a product of two binomials. This technique is especially useful when the quadratic can be easily broken down into simple factors. To factor a quadratic equation, it typically needs to be in the form \( ax^2 + bx + c = 0 \), where you'll be looking to break it down into the form \((px + q)(rx + s) = 0\).

This method is efficient because it allows us to find the roots of the equation by setting each factor equal to zero:

• If \((px + q) = 0\), then solve for \(x\).
• If \((rx + s) = 0\), solve for \(x\) again.

In some cases, the factors are not obvious, particularly if the quadratic does not factorize neatly into rational numbers. However, when the equation is factorable, it offers a quick solution method.
This is particularly clear in Equation D from our exercise example, where it was already presented in factored form as \((3x+1)(x-7)\). This allows for an immediate solution by solving each binomial factor.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula is particularly powerful because it can be used even when the quadratic cannot be easily factored. Here’s how each component functions:

• \(b\) represents the coefficient in front of the linear term \(x\).
• \(a\) is the coefficient of \(x^2\).
• \(c\) is the constant or the term without \(x\).
• The expression under the square root \( \sqrt{b^2 - 4ac} \) is known as the discriminant. It indicates the nature of the roots.

Depending on the value of the discriminant:

• If \(b^2 - 4ac > 0\), the quadratic has two distinct real solutions.
• If \(b^2 - 4ac = 0\), the quadratic has exactly one real solution.
• If \(b^2 - 4ac < 0\), the quadratic has two complex solutions.

Using the quadratic formula guarantees a solution, as seen in possibly Equation A, which couldn't be immediately factored but could be solved with this formula.

Standard Form of Quadratics

The standard form of a quadratic equation is typically expressed as \( ax^2 + bx + c = 0 \). Understanding and identifying this form is crucial for applying various solving techniques, such as factoring or using the quadratic formula.

To convert any equation into standard form, you need to ensure all terms are on one side of the equation, set equal to zero, which sometimes requires rearranging and simplifying terms. Let's look at an example.
Equation C was originally given as \( x^2 + x = 12 \). By rearranging to \( x^2 + x - 12 = 0 \), we convert it into the standard form, making it easier to identify coefficients:

• \(a = 1\)
• \(b = 1\)
• \(c = -12\)

Similarly, Equation A \( 3x^2 - 17x - 6 = 0 \) is already in standard form, where \( a = 3 \), \( b = -17 \), and \( c = -6 \). Converting equations into this form is a foundational step before applying other techniques to find the solutions.