Problem 21 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$x(2 x-1)=-3$$

Step-by-Step Solution

Answer

There are no real roots because the discriminant is negative.

Step 1: Expand the Equation

Start by expanding the equation \(x(2x - 1) = -3\). Distribute the \(x\) on the left side to get the quadratic equation. This results in \(2x^2 - x = -3\).

Step 2: Rearrange into Standard Form

Rewrite the equation in the standard quadratic form \(ax^2 + bx + c = 0\). Move \(-3\) to the left side to get \(2x^2 - x + 3 = 0\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -1\), and \(c = 3\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\), which is \((-1)^2 - 4 \times 2 \times 3 = 1 - 24 = -23\).

Step 5: Analyze the Discriminant

Since the discriminant is negative (\(-23\)), there are no real roots for the equation.

Key Concepts

Discriminant Analysis

To understand the nature of the roots of a quadratic equation, we perform discriminant analysis. A discriminant is denoted as the expression inside the square root of the quadratic formula, given by \( b^2 - 4ac \).
It serves as a critical indicator of the types of roots you can expect:

• If the discriminant is positive (greater than zero), the quadratic equation has two distinct real roots. This means the parabola will intersect the x-axis at two points.
• If the discriminant equals zero, there is exactly one real root, indicating the parabola only grazes the x-axis.
• If the discriminant is negative, as in the exercise provided (where the discriminant is \(-23\)), there are no real roots. In such cases, the parabola does not touch the x-axis at any point but instead rests completely above or below it.

Understanding this tool helps anticipate whether solving will yield real solutions, imaginary solutions, or a single touching point.

Quadratic Formula

The quadratic formula is a versatile tool for solving quadratic equations. These equations follow the structure \( ax^2 + bx + c = 0 \).
The formula itself is\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
Here's how it works:

• Substitute the coefficients: Identify the coefficients \( a \), \( b \), and \( c \) from your equation. In the example given, \( a = 2 \), \( b = -1 \), and \( c = 3 \).
• Compute the discriminant: This involves the expression \( b^2 - 4ac \). It's used to determine the nature of the roots, as shown in the discriminant analysis.
• Solving for \( x \): Use the results from the discriminant to solve. If the discriminant is non-negative, plug back into the formula to find your roots.

The quadratic formula is applicable to every quadratic equation, making it an essential method in algebra.

Real Roots

A real root of a quadratic equation is a solution where the function equals zero at certain values of \( x \). The concept of real roots ties back to the roots on a graph where the parabola intersects the x-axis.
Real roots can appear in different forms:

• Two distinct real roots: This occurs when the discriminant is positive. Each real root represents a unique intersection point on the x-axis.
• One real root: When the discriminant is zero, the parabola just touches the x-axis at one point, making it a double root.
• No real roots: As in the presented exercise, the negative discriminant indicates that no real values of \( x \) will meet the condition of equal zero since the parabola does not touch the x-axis. Such cases often yield complex or imaginary roots instead of real solutions.

Understanding real roots is crucial in determining where quadratic equations find their solutions on a graph.