Problem 41 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ z^{2}-\frac{3}{2} z+\frac{9}{16}=0 $$

Step-by-Step Solution

Answer

The real solution is \(z = \frac{3}{4}\).

Step 1: Identify the Quadratic Equation

The equation given is \(z^2 - \frac{3}{2}z + \frac{9}{16} = 0\). This is in the standard form of a quadratic equation, \(az^2 + bz + c = 0\), where \(a = 1\), \(b = -\frac{3}{2}\), and \(c = \frac{9}{16}\).

Step 2: Solve using the Quadratic Formula

The quadratic formula is \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plug in the values of \(a\), \(b\), and \(c\):\[z = \frac{-(-\frac{3}{2}) \pm \sqrt{(-\frac{3}{2})^2 - 4 \cdot 1 \cdot \frac{9}{16}}}{2 \cdot 1}\].

Step 3: Simplify Inside the Square Root

Calculate \((-\frac{3}{2})^2\), which equals \(\frac{9}{4}\). Then calculate \(4 \cdot 1 \cdot \frac{9}{16} = \frac{9}{4}\). Substitute these into the formula: \(z = \frac{\frac{3}{2} \pm \sqrt{\frac{9}{4} - \frac{9}{4}}}{2}\).

Step 4: Evaluate the Discriminant

The discriminant is \(b^2 - 4ac\). We have \(\frac{9}{4} - \frac{9}{4} = 0\). This means the discriminant is zero, indicating there is exactly one real solution.

Step 5: Calculate the Solution

Since the discriminant is zero, \(z = \frac{\frac{3}{2}}{2} = \frac{3}{4}\). This is the only real solution to the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra, used especially to find solutions to quadratic equations of the form \( ax^2 + bx + c = 0 \). This powerful formula is expressed as:\[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a\), \(b\), and \(c\) are coefficients of the quadratic equation, which you plug into the formula to solve for \(z\). It provides a straightforward way to find the roots of a quadratic equation, even when factoring is not possible.
The formula works by calculating values that balance the quadratic equation to zero using operations of addition, subtraction, division, and finding square roots. It is essential to correctly identify \(a\), \(b\), and \(c\) from the given quadratic equation before using this formula. Make sure to practice substituting these values accurately for consistent results.

Discriminant

Within the quadratic formula, a critical component is the discriminant, denoted by \(b^2 - 4ac\). This is the part under the square root and plays a pivotal role in determining the nature of the solutions:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, indicating the parabola touches the x-axis at one point.
• If the discriminant is negative, there are no real solutions, but rather two complex solutions.

In our exercise, the discriminant was calculated as zero, leading to a single real solution. Understanding the discriminant helps students predict the type and number of solutions without fully solving the equation.
This knowledge also aids in graphing quadratic functions, signaling whether and how they intersect the x-axis.

Real Solutions

Real solutions to a quadratic equation are the points where the graph of the equation intersects the x-axis. In many practical scenarios, like projectile motion or physics, finding these solutions is essential. They represent tangible, usable values in equations modeling real-world situations.
Real solutions are influenced directly by the discriminant:- When the discriminant is zero, as so in our given equation \(z^2 - \frac{3}{2}z + \frac{9}{16} = 0\), the equation has one real solution, \(z = \frac{3}{4}\).This means the vertex of the parabola represented by this quadratic equation precisely touches the x-axis. In general, if a quadratic has real solutions, these can be graphically seen as the x-intercepts of its parabola. By evaluating the discriminant first, you can understand the solutions before proceeding to solve them.

Completing the Square

Completing the square is another method for solving quadratic equations and can also provide insights into the nature of a quadratic expression. It involves rearranging the equation so that one side forms a perfect square trinomial. This technique is helpful when analyzing the vertex form of a quadratic, providing insights into its graph.Let's briefly discuss how this works:1. Start with the standard form of the equation \(ax^2 + bx + c = 0\).2. Divide all terms by \(a\) if \(a eq 1\).3. Rearrange to isolate the constant \(c\) on one side.4. Take half of the coefficient of \(b\), square it, and add this square to both sides.5. Factor to form a perfect square trinomial on one side and simplify the other.For the equation \(z^2 - \frac{3}{2}z + \frac{9}{16} = 0\), completing the square isn't necessary, as solving the quadratic formula directly revealed the solution. However, understanding this process is valuable, especially in transforming and understanding the structure and vertex of the equation graphically.