Problem 32 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. (Lesson 9.7) $$ 8 x^{2}-8 x+2=0 $$

Step-by-Step Solution

Answer

The quadratic equation \(8x^{2} - 8x + 2 = 0\) has one real solution.

Step 1: Identify the coefficients

From the given equation \(8x^{2} - 8x + 2 = 0\), we have a = 8, b = -8, and c = 2.

Step 2: Calculate the discriminant

The discriminant formula is \(b^{2} - 4ac\). Substituting a = 8, b = -8, and c = 2 into the formula, we get \((-8)^{2} - 4*8*2 = 64 - 64 = 0\).

Step 3: Analyze the result

Since the calculated discriminant equals to zero, this indicates that our quadratic equation has exactly one real solution.

Key Concepts

Understanding Discriminant Analysis

In the realm of quadratic equations, the discriminant provides critical information about the nature of the solutions to the equation. By employing the formula \( b^2 - 4ac \) where a, b, and c are coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), one can determine the type and number of solutions.

The discriminant can lead to three possible scenarios:

• If the discriminant is positive, we get two distinct real solutions.
• A discriminant of zero indicates exactly one real solution, which is also a repeated or double root.
• A negative discriminant suggests there are no real solutions; rather, there are two complex solutions, which are conjugates of each other.

In the case of the given equation \( 8x^2 - 8x + 2 = 0 \), the discriminant turns out to be zero, implying that there is one unique real solution to the equation.

The Quadratic Formula and Its Use

Solving quadratic equations is often simplified by using the quadratic formula, which is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). This formula provides a straightforward path to finding the equation's solutions by substituting the values of a, b, and c from the equation into the formula.

To apply the quadratic formula correctly, one should first ensure the equation is in standard form and then identify the coefficients clearly. For example, in our equation \( 8x^2 - 8x + 2 = 0 \), upon calculating the discriminant as zero, we can anticipate a single real solution. This sole solution can be found by placing the coefficients into the quadratic formula. Since the discriminant is zero, the formula simplifies to \( x = \frac{{-(-8)}}{{2 \cdot 8}} \) which equals \( x = \frac{1}{2} \).

Examining the Real Solutions

Real solutions to quadratic equations are the x-intercepts, or roots, where the graph of the quadratic function intersects the x-axis. The nature of these solutions is directly tied to the discriminant value computed from the quadratic equation. With a positive discriminant, the graph crosses the x-axis at two distinct points; with zero, it touches the axis at one point; and with a negative discriminant, it does not intersect the x-axis at all.

When dealing with real solutions, it's important to recognize that a discriminant equal to zero not only means one real solution but also that this solution is where the vertex of the parabola is situated on the x-axis. In the context of our equation, we have determined a single real solution at \( x = \frac{1}{2} \), which would be the turning point of the parabola, showcasing the symmetrical nature of the graph of a quadratic function.