Problem 38 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Equation

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

Step 2: Use the Quadratic Formula

To find the roots of the quadratic equation \( ax^2 + bx + c = 0 \), use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug in the values: \( a = 1 \), \( b = -\frac{3}{2} \), and \( c = \frac{9}{16} \).

Step 3: Calculate the Discriminant

Compute the discriminant \( D = b^2 - 4ac \). Here, \( b^2 = \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \) and \( 4ac = 4 \cdot 1 \cdot \frac{9}{16} = \frac{36}{16} = \frac{9}{4} \). Thus, \( D = \frac{9}{4} - \frac{9}{4} = 0 \).

Step 4: Find the Solution

Since the discriminant \( D = 0 \), there is exactly one real solution (a double root) given by \( \theta = \frac{-b}{2a} \). Substituting \( b = -\frac{3}{2} \) and \( a = 1 \) into the formula gives \( \theta = \frac{3/2}{2} = \frac{3}{4} \).

Step 5: Verify the Solution

Substitute \( \theta = \frac{3}{4} \) back into the original equation: \( \left(\frac{3}{4}\right)^2 - \frac{3}{2} \cdot \frac{3}{4} + \frac{9}{16} = \frac{9}{16} - \frac{9}{8} + \frac{9}{16} = 0 \). This confirms that \( \theta = \frac{3}{4} \) is indeed a solution.

Key Concepts

Quadratic Formula

The quadratic formula is essential when solving quadratic equations, which have the general form: \[ ax^2 + bx + c = 0 \] This formula provides a simple way to determine the solutions (or roots) of such equations: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are coefficients in the quadratic equation. This formula is powerful because it can be applied to any quadratic equation, revealing all possible solutions. The plus-minus (±) sign indicates that there might be two different solutions—one obtained using the plus symbol and the other using the minus symbol in the formula.

Discriminant

The discriminant is a crucial part of the quadratic formula. It is the expression under the square root sign: \[ b^2 - 4ac \] This value can tell you a lot about the nature of the roots without having to compute them exactly:

• If the discriminant is greater than zero, the equation has two distinct real solutions.
• If it is exactly zero, there is exactly one real solution, known as a double root.
• If the discriminant is less than zero, there are no real solutions, but there are two complex solutions.

Thus, the discriminant helps quickly assess whether the solutions are real or complex.

Real Solutions

In the context of quadratic equations, real solutions are the values of \(x\) that satisfy the equation: \[ ax^2 + bx + c = 0 \] These solutions are real numbers that you can plot on a number line. Determining the real solutions involves using the quadratic formula and evaluating the discriminant. When the discriminant is non-negative (zero or positive), the quadratic equation has real solutions. This distinction is practical for understanding if the roots can be measured or visualized using standard real numbers.

Double Root

A double root, also known as a repeated or coincident root, arises when the discriminant \(b^2 - 4ac\) of a quadratic equation is zero. This means that the quadratic equation has exactly one unique solution or root, occurring twice in the context of its algebraic multiplicity. Mathematically, this happens when: \[ x = \frac{-b}{2a} \] This value is a single point on the number line where the parabola represented by the quadratic equation just touches the \(x\)-axis. Solving for a double root is straightforward, as the calculations involve simpler arithmetic without the square root component in the quadratic formula.