Problem 53 - Algebra for College Students

Question

Why must we change the equation \(3 x^{2}-2 x=4\) to \(3 x^{2}-\) \(2 x-4=0\) before applying the quadratic formula? The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth.

Step-by-Step Solution

Answer

Change to standard form to apply the quadratic formula.

Step 1: Understand the Quadratic Equation

A quadratic equation is typically in the form \(ax^2 + bx + c = 0\). This standard form is crucial when using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 2: Transform Given Equation

Given the equation \(3x^2 - 2x = 4\), move all terms to one side of the equation to set it to 0, transforming it into \(3x^2 - 2x - 4 = 0\).

Step 3: Set Up for the Quadratic Formula

The equation must be in the standard form \(ax^2 + bx + c = 0\) for the quadratic formula to be applied correctly. Only then can you identify coefficients: \(a = 3\), \(b = -2\), and \(c = -4\).

Key Concepts

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations. It's used to find the roots or solutions of any quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). This formula states that the solutions for \(x\) are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This equation might look complex at first glance. However, it's just a method to find where the graph of the quadratic function will intersect the x-axis. The "\(\pm\)" symbol means there are two possible solutions: one using the plus and one using the minus.

• "\(b^2 - 4ac\)" is called the discriminant and it determines the nature of the roots – whether they are real or complex.
• Use this formula when you cannot easily factor the quadratic equation.

In the quadratic formula, each letter \(a\), \(b\), and \(c\) corresponds to the coefficients in the standard form of a quadratic equation. It's a straightforward way to solve quadratics when other methods are too complicated.

Standard Form

The standard form of a quadratic equation is crucial for applying the quadratic formula. A quadratic equation is in its standard form when it is written as:
\[ax^2 + bx + c = 0\]
In this setup:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Each term must be on one side of the equation resulting in 0 on the other. This is important because:

• The coefficients \(a\), \(b\), and \(c\) are directly used in the quadratic formula.
• It allows for easy substitution into the formula to solve for \(x\).

In the original exercise, transforming \(3x^2 - 2x = 4\) to the standard form \(3x^2 - 2x - 4 = 0\), makes it possible to identify \(a = 3\), \(b = -2\), and \(c = -4\) easily, which are needed for the formula. Writing equations in this format is necessary for consistent and accurate problem-solving.

Solving Quadratic Equations

Solving quadratic equations can be approached through various methods, such as:

• Factoring the quadratic expression, if possible.
• Completing the square to find the roots.
• Using the quadratic formula as a foolproof option.

For the equation \(3x^2 - 2x - 4 = 0\), the quadratic formula is the most efficient method. To apply it:
1. Substitute \(a = 3\), \(b = -2\), and \(c = -4\) into the quadratic formula.2. Calculate the discriminant \(b^2 - 4ac\). In this case, it determines whether you get real or complex roots.3. Compute the solutions using the formula to find the values of \(x\) that satisfy the equation.
The problem provided examples of exact solutions in a complex form \(2 \pm \sqrt{41}\), and rational approximations to the nearest one-thousandth. This demonstrates how the formula can precisely find the x-values where a quadratic equation hits the x-axis, representing the real roots.