Problem 30 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. (Lesson 9.7) $$ x^{2}-2 x+4=0 $$

Step-by-Step Solution

Answer

The equation has no real solution.

Step 1: Identify coefficients a, b, and c

From \( x^{2}-2x+4 = 0 \), we get a = 1, b = -2, and c = 4.

Step 2: Compute the Discriminant D

Substitute a, b, and c into the Discriminant formula: \( D = b^2-4ac = (-2)^2 - 4*1*4 = 4-16 = -12

Step 3: Conclude The Number of Solutions

Since D < 0 (D= -12), it implies the equation has no real solution.

Key Concepts

Discriminant

The discriminant is a crucial part of solving quadratic equations. It helps us determine the nature of the solutions without actually solving the equation. The discriminant, denoted as \( D \), is calculated using the formula \( D = b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

The value of the discriminant reveals a lot about the quadratic equation's solutions:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, also known as a double root.
• If \( D < 0 \), there are no real solutions; the solutions are complex or imaginary.

In our exercise, the discriminant \( D = -12 \), which means the equation \( x^2 - 2x + 4 = 0 \) has no real solutions.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation \( ax^2 + bx + c = 0 \). These solutions are called "real" because they can be plotted on the real number line, as opposed to imaginary solutions.

The nature of real solutions is directly related to the discriminant \( D \):

• Two distinct real solutions exist when \( D > 0 \).
• One real solution occurs when \( D = 0 \), indicating the vertex of the parabola touches the x-axis.
• No real solutions are present when \( D < 0 \). In this case, the parabola does not intersect the x-axis at all.

In the example from the exercise, since the discriminant is negative, no real values of \( x \) satisfy the equation. Therefore, the parabola described by \( x^2 - 2x + 4 = 0 \) does not touch or cross the x-axis.

Coefficients

Coefficients are the numerical or constant factors in the terms of a quadratic equation. In the standard form \( ax^2 + bx + c = 0 \), coefficients \( a \), \( b \), and \( c \) have specific roles:

• \( a \): the coefficient of \( x^2 \), known as the quadratic coefficient, which affects the parabola's direction and width.
• \( b \): the coefficient of \( x \), known as the linear coefficient, influencing the parabola's tilt and axis of symmetry.
• \( c \): the constant term, affecting the vertical position of the parabola on the graph.

Identifying these coefficients is the first step in many algebraic procedures, such as calculating the discriminant or finding the vertex. For instance, in the equation \( x^2 - 2x + 4 = 0 \), we identified \( a = 1 \), \( b = -2 \), and \( c = 4 \). Knowing these values allows us to compute the discriminant and understand the nature of the solutions.

Algebraic Solutions

Solving quadratic equations algebraically involves finding the values of \( x \) that make the equation \( ax^2 + bx + c = 0 \) true. There are several approaches, including factoring, using the quadratic formula, or completing the square.

1. **Factoring**: This involves rewriting the quadratic expression as a product of simpler expressions. This is only possible when the roots are rational numbers.
2. **Quadratic Formula**: For all types of solutions, this formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), gives the solutions directly using the coefficients \( a \), \( b \), and \( c \).
3. **Completing the Square**: This transforms the equation into a perfect square trinomial, making it easier to solve for \( x \).

Each method has its strengths and applicability depending on the equation's characteristics. However, the discriminant helps determine which methods might be feasible—especially when the solutions are complex, as with our example \( x^2 - 2x + 4 = 0 \), where no real solutions exist due to a negative discriminant.