Problem 71 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ 2 x(x-2)=x-4 $$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Expanding the Left Side

Begin by expanding the left side of the equation. Distribute the 2x across the binomial \((x - 2)\). This gives us: \[ 2x(x - 2) = 2x^2 - 4x \] This transforms the original equation into: \[ 2x^2 - 4x = x - 4 \]

Step 2: Re-arranging the Equation

To solve for \(x\), we need to bring all terms to one side. Subtract \(x\) and add 4 to both sides to get everything on the left: \[ 2x^2 - 4x - x + 4 = 0 \] Simplifying, this becomes: \[ 2x^2 - 5x + 4 = 0 \]

Step 3: Using the Quadratic Formula

The equation is now in standard form \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = -5\), and \(c = 4\). Apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values to find \(x\): \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \times 2 \times 4}}{2 \times 2} \]

Step 4: Calculating the Discriminant

First, compute the discriminant: \[ b^2 - 4ac = (-5)^2 - 4 \times 2 \times 4 = 25 - 32 = -7 \] Since the discriminant is negative, the quadratic equation has no real solutions.

Key Concepts

Discriminant

In the context of quadratic equations, the discriminant is a key concept that helps determine the nature of the solutions. It is part of the quadratic formula, and its value influences whether solutions are real or complex. Specifically, the discriminant is calculated using the formula:
\[ b^2 - 4ac \]
Here, \(b\), \(a\), and \(c\) are coefficients present in the quadratic equation in standard form. The discriminant can help us understand:

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If the discriminant is zero, the equation has exactly one real solution, meaning both solutions are the same (a double root).
• Finally, if the discriminant is negative, the quadratic equation has no real solutions. Instead, it has two complex conjugate solutions.

In the example problem, the discriminant is calculated as \( (-5)^2 - 4 \times 2 \times 4 \), which equals \(-7\). Since \(-7\) is negative, the equation has no real solutions, confirming only complex solutions exist.

Standard Form

The standard form of a quadratic equation is a way of writing it clearly and consistently. It is represented as:
\[ ax^2 + bx + c = 0 \]
Where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This form is essential as it allows for easy identification of the coefficients necessary for various solving methods, such as the quadratic formula and the discriminant. To convert any quadratic equation into standard form, follow these steps:

• Distribute any multiplied terms across the parentheses when needed, as demonstrated in the first step of our solution.
• Rearrange the terms by ensuring that all variable terms are on one side of the equation, resulting in zero on the opposite side.
• Combine like terms to simplify the equation fully before proceeding to further analysis.

In the given problem, after expanding and rearranging, the equation was set to \(2x^2 - 5x + 4 = 0\), thereby achieving the standard form.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of quadratic equations, especially when factoring is not feasible. The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula can solve any quadratic equation, whether it has real or complex solutions. Here's how it works:

• \(-b\) changes the sign of \(b\) to assist in the calculation of the solution.
• The discriminant \(\sqrt{b^2 - 4ac}\) is essential in determining the nature of the roots, as explained previously.
• The entire expression is divided by \(2a\), ensuring the solution accounts for the squared term coefficient.

In our example, upon substituting \(a = 2\), \(b = -5\), and \(c = 4\) into the formula, the negative discriminant of \(-7\) revealed that real solutions do not exist. As a result, this iteration of the formula shows the insight into the solutions being complex.