Problem 23 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(2 x^{2}-4 x+3=0\)

Step-by-Step Solution

Answer

The given equation \(2 x^{2}-4x+3=0\) has no real solutions.

Step 1: Identify a, b, and c from the equation

In the given quadratic equation \(2 x^{2}-4x+3=0\), the value of 'a' (the coefficient of \(x^{2}\)) is 2, 'b' (the coefficient of 'x') is -4, and 'c' (the constant) is 3.

Step 2: Calculate the discriminant

Put the identified coefficients into the discriminant formula \[D = b^{2}-4ac\]. So, \( D = (-4)^{2} - 4*2*3 = 16 - 24 = -8 \]

Step 3: Determine the number of solutions

Based on the result of the discriminant D we've got, which is -8, it can be stated that the given equation has no real solutions since D < 0.

Key Concepts

Understanding the Discriminant

The discriminant is a key component in solving quadratic equations. It provides insight into the nature and number of solutions, often before diving into more complex calculation steps. When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by the formula:

• \( D = b^2 - 4ac \)

The value of the discriminant can help determine whether the solutions to the quadratic equation are real or complex numbers.
For a better understanding:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, often called a repeated or double root.
• If \( D < 0 \), there are no real solutions, only complex numbers involve imaginary parts.

In our example (\(2x^2 - 4x + 3 = 0\)), where \(D = -8\), the discriminant is negative. This tells us that the quadratic equation does not have real solutions, which emphasizes the importance of thoroughly understanding the discriminant's role.

Identifying Real Solutions

Real solutions in the context of quadratic equations are those that exist on the real number line. Understanding whether a quadratic equation has real solutions or not can save you a lot of time.
Once the discriminant is calculated, its sign (positive, zero, or negative) determines the possibility of real solutions. As mentioned, the discriminant being negative, as in our example \(D = -8\), suggests no real solutions exist and indicates complex solutions instead.
Real solutions are desirable in many practical applications as they provide meaningful answers in real-world scenarios such as physics, engineering, and biology. Therefore, understanding whether solutions are real or not is crucial.

• Real solutions appear when the parabola intersects the x-axis.
• When no real solutions are present, the parabola does not touch the x-axis, implying it lies entirely above or below the axis.

Knowing if you are working with real solutions helps determine the approach you will take in solving problems involving quadratics.

The Quadratic Formula and Its Usage

The quadratic formula is a powerful tool for finding the solutions of quadratic equations, particularly when factoring is not easily applicable. The quadratic formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides the solutions for \(x\) by utilizing the values of \(a\), \(b\), and \(c\) directly from the quadratic equation \(ax^2 + bx + c = 0\).
The "\(\pm\)" symbol indicates that there may be two solutions: one by using addition, and the other using subtraction. The term under the square root sign, \(b^2 - 4ac\), is the discriminant, and its impact is crucial:

• If \(D > 0\), two different real solutions result from the plus and minus operations.
• A \(D = 0\) yields one real solution since adding and subtracting zero yield the same value.
• With \(D < 0\), the square root of a negative number results in complex solutions, indicating no real roots.

The quadratic formula's universality allows it to be used in various situations, simplifying processes that might otherwise require more cumbersome methods. It is especially helpful in confirming the type and number of solutions indicated by the discriminant.