Problem 26 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

In the equation \(x^{2}-2 x+4=0\), coefficients are: \(a = 1\), \(b = -2\), \(c = 4\)

Step 2: Calculate the discriminant

Use the formula \(D = b^{2} - 4ac\) , then substitute the values a, b and c into the equation: \(D = (-2)^{2} - 4*1*4 = 4 - 16 = -12\)

Step 3: Determine the number of solutions

As the discriminant D = -12 is less than 0, there are no real solutions for this quadratic equation

Key Concepts

Discriminant

The discriminant is a crucial part of solving quadratic equations. It gives us insight into the nature of the solutions without having to solve the equation completely. The discriminant is found using the formula:

• \( D = b^2 - 4ac \)

Where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation in the form \( ax^2 + bx + c = 0 \).
The value of the discriminant helps us determine the solution:

• If \( D > 0 \), the quadratic equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution (also called a repeated or double root).
• If \( D < 0 \), there are no real solutions, but two complex ones.

This concept is particularly useful because it allows us to understand how many solutions exist at a glance by only looking at the calculated value of \( D \). In our exercise, since the discriminant is \( -12 \) (less than zero), the quadratic equation has no real solutions.

Real Solutions

Real solutions to quadratic equations are the values that satisfy the equation when solved under real numbers. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), solutions are found using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

Here, \( D \) is the discriminant, \( b^2 - 4ac \).
To have real solutions, the discriminant \( D \) must be zero or positive. When \( D > 0 \), the formula results in two distinct real solutions because the square root of a positive number is a real number. If \( D = 0 \), the solution simplifies to \( x = \frac{-b}{2a} \), providing a single real solution since the \( \sqrt{0} = 0 \).
However, in our specific problem, \( D = -12 \), which results in no real solutions. The presence of a negative discriminant means the equation's solutions will involve the square root of a negative number, leading to complex solutions involving imaginary numbers.

Coefficients

Coefficients are the numerical or constant parts of the terms in a polynomial, and they play a pivotal role in solving quadratic equations. In the standard quadratic form \( ax^2 + bx + c = 0 \), the coefficients are labeled as:

• \( a \) - the coefficient of \( x^2 \)
• \( b \) - the coefficient of \( x \)
• \( c \) - the constant term

Identifying these coefficients is the first step in many solution methods for quadratic equations, including calculating the discriminant. For our exercise:

• \( a = 1 \)
• \( b = -2 \)
• \( c = 4 \)

These coefficients are plugged into the discriminant formula \( b^2 - 4ac \) to evaluate the nature of the solutions. The values defined will determine the number and type of solutions for the equation, showcasing their fundamental importance in problem-solving.