Problem 33 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(5 x^{2}+4 x+\frac{4}{5}=0\)

Step-by-Step Solution

Answer

After calculating the discriminant, based on its value, the equation will be determined as having two real solutions, one real solution, or no real solution.

Step 1: Identify the coefficients

In the equation \(5x^2 + 4x + \frac{4}{5} = 0\), the coefficients are \(a = 5\), \(b = 4\), and \(c = \frac{4}{5}\).

Step 2: Calculate the discriminant

The discriminant \(D\) is computed as \(D = b^2 - 4ac = 4^2 - 4 * 5 * \frac{4}{5}\).

Step 3: Determine the nature of the roots

If the discriminant is positive, the equation will have two distinct real roots. If it is zero, the equation will have exactly one real root. If the discriminant is negative, there are no real solutions.

Key Concepts

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation provides vital information about the nature of its roots without actually solving the equation. Given a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\).

Let's apply this to our exercise. With the coefficients \(a = 5\), \(b = 4\), and \(c = \frac{4}{5}\), we calculate the discriminant as follows: \[D = b^2 - 4ac = 4^2 - 4 \times 5 \times \frac{4}{5} = 16 - 16 = 0\]. The value of the discriminant in this case is zero, which leads us to an important conclusion about the nature of the roots, as we will explore further below.

Understanding the discriminant is essential for students as it not only helps in determining the number of roots but also their nature — whether they are real, repeated, or complex. Remember, if you ever get a positive discriminant, your equation will have two distinct real roots; if it’s zero, there will be one real repeated root; and a negative discriminant indicates that you'll have two complex roots.

Nature of Quadratic Equation Roots

Once the discriminant has been found, determining the nature of the roots becomes straightforward. In our exercise example, since the discriminant \(D = 0\), we know that the quadratic equation will have exactly one real root. This is because a zero discriminant indicates that the quadratic equation's graph touches the x-axis at a single point, representing the repeated root.

Let's recollect the rules: If the discriminant \(D > 0\), the quadratic equation has two distinct real roots. When \(D = 0\), there is one real root, which can also be referred to as a repeated or double root. On the other hand, if \(D < 0\), the roots are not real but complex and they come in a conjugate pair.

These concepts serve as a powerful tool for students to quickly assess the outcome of a quadratic equation without extensive calculations. It also helps in understanding the graph of a quadratic equation. Remember, the roots are the x-values where the quadratic graph intersects the x-axis, so their nature also reflects the shape and position of the parabola formed by the equation.

Identifying Coefficients in Quadratic Equations

Properly identifying coefficients in a quadratic equation is the first step toward understanding and solving it. A standard quadratic equation is of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known as the coefficients. Here, \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) represents the constant term.

In our example, \(5x^2 + 4x + \frac{4}{5} = 0\), the coefficients are clearly \(a = 5\), \(b = 4\), and \(c = \frac{4}{5}\). Understanding these allows students to navigate through subsequent steps like calculating the discriminant or factoring the equation if possible.

It's crucial to note that the value of \(a\) will affect the direction in which the parabola opens. If \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward. These details not only assist in solving equations but also enhance comprehension of quadratic functions as a whole. Being comfortable with identifying coefficients is key to mastering many aspects of quadratic equations and builds a solid foundation for further concepts in algebra.