Problem 32 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(\frac{1}{4} x^{2}-2 x+4=0\)

Step-by-Step Solution

Answer

The equation has one real solution.

Step 1: Identify a, b, and c

For the equation \(\frac{1}{4} x^{2}-2 x+4=0\), a is \(\frac{1}{4}\), b is -2, and c is 4.

Step 2: Calculate the Discriminant

The discriminant, \(D\), is calculated using the formula \(b^2-4ac\). Substituting the given values, \(D = (-2)^2 - 4 * \(\frac{1}{4}\) * 4 = 4 - 4 = 0\).

Step 3: Determine Number of Solutions

Since the discriminant is 0, the equation has exactly one real solution.

Key Concepts

Discriminant in Quadratics

The discriminant is a key concept in quadratic equations, as it helps us determine the nature of the solutions without actually solving the equation. The discriminant, denoted by the symbol 'D', is part of the quadratic formula and is calculated as \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the standard quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant can lead to three different scenarios:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution.
• If \(D < 0\), there are no real solutions, but two complex solutions.

Understanding the discriminant is essential for predicting solution outcomes and forms the basis for solving quadratic equations efficiently.

Quadratic Formula

The quadratic formula is a widely used method for finding the solutions of a quadratic equation. Presented as \(x = \frac{-b \pm \sqrt{D}}{2a}\), this formula provides the solution(s) directly based on the coefficients \(a\), \(b\), and \(c\) of the equation \(ax^2 + bx + c = 0\). The symbol \(\pm\) indicates that there are generally two solutions — based on the positive and negative square roots of the discriminant.

By plugging in the values of the coefficients and following the arithmetic operations, one can discern the real solutions of the quadratic equation. The value under the square root in the formula (\(\sqrt{D}\)) is essentially the discriminant, emphasizing its importance in solving these equations. Clear comprehension of the quadratic formula is crucial for students in algebra.

Number of Real Solutions

The number of real solutions to a quadratic equation relates directly to the value of the discriminant. Knowing whether there are two, one, or no real solutions helps us grasp the function's behavior without graphing it. The discriminant's value serves as the indicator:

• When \(D > 0\), the quadratic equation intersects the x-axis at two points, signifying two unique real solutions.
• When \(D = 0\), the parabola touches the x-axis at one point only, representing a single real solution — sometimes referred to as a repeated or double root.
• When \(D < 0\), the parabola does not touch the x-axis at all, indicating there are no real solutions but instead complex ones.

For the given equation \(\frac{1}{4} x^{2}-2 x+4=0\), with \(D = 0\), we conclude that it has one real solution. This concept is vital for students as it connects the algebraic sign of the discriminant to the geometric interpretation of a quadratic graph.