Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ \frac{3}{2} z^{2}-4 z-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( z = \frac{4 + \sqrt{22}}{3} \) and \( z = \frac{4 - \sqrt{22}}{3} \).

Step 1: Identify the coefficients

The given quadratic equation is \( \frac{3}{2}z^2 - 4z - 1 = 0 \). Identify the coefficients: \( a = \frac{3}{2} \), \( b = -4 \), and \( c = -1 \).

Step 2: Write the quadratic formula

The quadratic formula is given by \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the equation.

Step 3: Calculate the discriminant

The discriminant \( \Delta \) is calculated as \( b^2 - 4ac \). Substitute the values: \( (-4)^2 - 4\left(\frac{3}{2}\right)(-1) \). This simplifies to \( 16 + 6 = 22 \).

Step 4: Apply the quadratic formula

Substitute \( a = \frac{3}{2} \), \( b = -4 \), and \( c = -1 \) into the quadratic formula: \( z = \frac{-(-4) \pm \sqrt{22}}{2\left(\frac{3}{2}\right)} \). This simplifies to \( z = \frac{4 \pm \sqrt{22}}{3} \).

Step 5: Separate the two solutions

The solutions to the equation are \( z = \frac{4 + \sqrt{22}}{3} \) and \( z = \frac{4 - \sqrt{22}}{3} \). These are the two roots of the quadratic equation.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It is typically written in the standard form: \[ ax^2 + bx + c = 0 \] where:

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

Quadratic equations are crucial in algebra because they form the foundation for more complex concepts. They can represent various physical and geometric problems, making their solutions essential in mathematics. In the given problem, the quadratic equation is \( \frac{3}{2}z^2 - 4z - 1 = 0 \), featuring coefficients \( a = \frac{3}{2} \), \( b = -4 \), and \( c = -1 \). Identifying these coefficients correctly is the first step towards solving the equation using the quadratic formula.

Discriminant

The discriminant is a component of the quadratic formula, which helps determine the nature of the roots of a quadratic equation. It is denoted by the symbol \( \Delta \) and is calculated using the formula: \[ \Delta = b^2 - 4ac \] Notes on the discriminant:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root (or a repeated real root).
• If \( \Delta < 0 \), the equation has two complex roots (not real).

For the given equation, the discriminant is calculated as: \[ (-4)^2 - 4\left(\frac{3}{2}\right)(-1) \] This simplifies to \( 16 + 6 = 22 \). Since \( 22 > 0 \), the quadratic equation has two distinct real roots.

Roots of the Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation, making the expression equal to zero. We can find these roots using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is a dependable method for finding the solutions or roots of the quadratic equation when factorization is complex or impossible.For the quadratic equation \( \frac{3}{2}z^2 - 4z - 1 = 0 \), substituting the coefficients \( a = \frac{3}{2} \), \( b = -4 \), and \( c = -1 \) into the formula yields: \[ z = \frac{-(-4) \pm \sqrt{22}}{2\left(\frac{3}{2}\right)} \] This further simplifies to: \[ z = \frac{4 \pm \sqrt{22}}{3} \] Thus, the two solutions are:

• \( z = \frac{4 + \sqrt{22}}{3} \)
• \( z = \frac{4 - \sqrt{22}}{3} \)

These are the roots of the quadratic equation, providing the points at which the curve intersects the \( z \)-axis in a coordinate system.