Problem 55 - Algebra and Trigonometry

Question

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-2.450 x+1.501=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions because the discriminant is negative.

Step 1: Identifying the coefficients

In the quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients \( a \), \( b \), and \( c \). For the equation \( x^2 - 2.450x + 1.501 = 0 \), we have \( a = 1 \), \( b = -2.450 \), and \( c = 1.501 \).

Step 2: Using the quadratic formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the identified coefficients: \( a = 1 \), \( b = -2.450 \), \( c = 1.501 \) into this formula.

Step 3: Calculating the discriminant

The discriminant \( \Delta \) is \( b^2 - 4ac \). Substitute the values \( b = -2.450 \), \( a = 1 \), \( c = 1.501 \): \[ (-2.450)^2 - 4 \times 1 \times 1.501 = 6.0025 - 6.004 = -0.0015 \].

Step 4: Evaluating the square root

Since the discriminant \( \Delta = -0.0015 \) is negative, the equation has no real solutions (as only real numbers can be square rooted to yield real solutions). This means, strictly speaking, there are no real number solutions.

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). In this equation, \( x \) represents an unknown variable, and \( a \), \( b \), and \( c \) are known values with \( a eq 0 \). Quadratic equations are fundamental in algebra and have the characteristic feature of a squared term.

Solving a quadratic equation can reveal vital information in various fields such as physics, engineering, and finance. The ultimate goal is to find the values of \( x \) that make the equation true, known as the solutions of the equation.

It's important to remember that every quadratic equation can potentially have two solutions, one solution, or even no real solution. This depends on the method used and the values of \( a \), \( b \), and \( c \).

Real Solutions

Real solutions in the context of quadratic equations refer to solutions that are real numbers—numbers that can be positioned on the number line. Sometimes, quadratic equations may yield complex or imaginary solutions if they don't intersect the x-axis when graphed.

If a quadratic equation has real solutions, they can be either:

• Distinct and consist of two different numbers.
• Coinciding, where both solutions are the same, resulting in a single unique solution.

To determine the nature of the solutions, we utilize the quadratic formula and mainly focus on the discriminant. Regardless of the method, real solutions are crucial for solving many real-world problems effectively.

Discriminant

The discriminant is an essential component of the quadratic formula, symbolized by \( \Delta \). It is found under the square root in the quadratic formula and is calculated using the expression \( b^2 - 4ac \). The value of the discriminant provides insight into the nature of the solutions of a quadratic equation.

Here's how the discriminant affects the solutions:

• If \( \Delta > 0 \), the quadratic equation has two distinct real solutions.
• If \( \Delta = 0 \), it has exactly one real solution, often referred to as a repeated or double root.
• If \( \Delta < 0 \), the quadratic equation has no real solutions. This indicates that the solutions are complex instead.

Let's look at our specific case: For the equation \( x^2 - 2.450x + 1.501 = 0 \), the discriminant is calculated as \( -0.0015 \), which is less than zero. Therefore, no real solutions exist for this particular quadratic equation.