Problem 40 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$\frac{1}{2} w^{2}-\frac{1}{3} w+\frac{1}{4}=0$$

Step-by-Step Solution

Answer

The discriminant is \(-\frac{1}{36}\) and there are no real solutions.

Step 1: Identify the coefficients

In the quadratic equation, we need to identify the coefficients a, b, and c. The given equation is \(\frac{1}{2} w^{2}-\frac{1}{3} w+\frac{1}{4}=0\). Here, \(a = \frac{1}{2}\), \(b = -\frac{1}{3}\), and \(c = \frac{1}{4}\).

Step 2: Compute the discriminant

The discriminant of a quadratic equation \(ax^{2} + bx + c = 0\) is given by \(b^{2} - 4ac\). Substitute the values of \(a, b, \) and \(c\): \[\left( -\frac{1}{3} \right)^{2} - 4 \left( \frac{1}{2} \right) \left( \frac{1}{4} \right) = \frac{1}{9} - 4 \cdot \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{9} - \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{9} - \frac{1}{8}= -\frac{1}{36} \quad \Rightarrow \boxed{-\frac{1}{36}}\]

Step 3: Determine the number of real solutions

The number of real solutions depends on the value of the discriminant: \(D = b^{2} - 4ac\). If \(D < 0\), there are no real solutions. If \(D = 0\), there is exactly one real solution. If \(D > 0\), there are two distinct real solutions. Here, \(-\frac{1}{36} < 0\), so the quadratic equation has no real solutions.

Key Concepts

quadratic formula

The quadratic formula is essential for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula is derived from completing the square on a general quadratic equation, and it provides a direct way to find the roots (solutions) of the equation. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here:

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

To solve a quadratic equation using the quadratic formula:

1. Identify the coefficients \(a\), \(b\), and \(c\) from the equation
2. Calculate the discriminant \(D = b^2 - 4ac\)
3. Substitute \(a\), \(b\), and the discriminant into the quadratic formula
4. Simplify under the square root (if necessary) and solve for \(x\)

The quadratic formula allows us to determine both real and complex solutions depending on the discriminant's value.

real solutions

The nature of the solutions to a quadratic equation depends heavily on the discriminant \(D = b^2 - 4ac\). The discriminant tells us about the nature of the roots:

• If \(D > 0\), the equation has two distinct real solutions
• If \(D = 0\), the equation has exactly one real solution (also known as a repeated root)
• If \(D < 0\), the equation has no real solutions, but two complex solutions

In the given equation \(\frac{1}{2}w^2 - \frac{1}{3}w + \frac{1}{4} = 0\), we found the discriminant to be \(-\frac{1}{36}\), which is less than zero. Therefore, there are no real solutions to this equation, only complex solutions. This insight is crucial for correctly interpreting the results of solving quadratic equations in various contexts.

coefficients in quadratic equations

Understanding the role of coefficients \(a\), \(b\), and \(c\) in quadratic equations is fundamental to solving and analyzing these types of equations:

• \(a\): The coefficient of \(x^2\). It affects the direction and width of the parabola represented by the quadratic equation. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
• \(b\): The coefficient of \(x\). It influences the position of the vertex of the parabola along the x-axis. It can also affect the location of the axis of symmetry of the parabola.
• \(c\): The constant term. It represents the y-intercept of the parabola, showing where the graph crosses the y-axis.

In the equation \(\frac{1}{2}w^2 - \frac{1}{3}w + \frac{1}{4} = 0\), the coefficients are \(a = \frac{1}{2}\), \(b = -\frac{1}{3}\), and \(c = \frac{1}{4}\). Recognizing these coefficients is the first step in applying the quadratic formula and understanding the nature of the equation's solutions. By identifying these values accurately, you set the foundation for successful problem-solving.