Problem 42 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$9-24 z+16 z^{2}=0$$

Step-by-Step Solution

Answer

The discriminant is 0, so there is 1 real solution.

Step 1: Identify the coefficients

Rewrite the quadratic equation in the standard form \( ax^2 + bx + c = 0 In the given equation, $ 16 z^{2} - 24 z + 9 = 0 You can identify: a = 16, b = -24, c = 9 \)

Step 2: Calculate the discriminant

To find the discriminant, use the formula: \( b^{2} - 4ac \) Substituting the identified coefficients: \((-24)^{2} - 4(16)(9) = 576 - 576 = 0\)

Step 3: Determine the number of real solutions

Check the value of the discriminant: If the discriminant is greater than 0, there are 2 real solutions. If it is equal to 0, there is 1 real solution. If it is less than 0, there are no real solutions. Since the discriminant is 0, there is exactly 1 real solution for the equation.

Key Concepts

Introduction to Quadratic Equations

A quadratic equation is a mathematical equation of the form \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are coefficients and \( a eq 0 \). Quadratic equations are called 'quadratic' because they deal with the square of the unknown variable, \( x^2 \).

Here’s a breakdown of the coefficients we typically see:

• 'a' is the leading coefficient and it multiplies the square term.
• 'b' is the coefficient of the linear term.
• 'c' is the constant term.

The standard form of the quadratic equation makes it easier to calculate important elements like the discriminant and to find the roots (solutions) of the equation. Common methods to solve quadratic equations include factoring, completing the square, and using the quadratic formula.

Understanding the Discriminant in Quadratic Equations

The discriminant is a key element in determining the nature of the solutions of a quadratic equation. To find it, you use the formula: \( b^2 - 4ac \).

The discriminant provides insights:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution.
• If \( b^2 - 4ac < 0 \), there are no real solutions (the solutions are complex numbers).

Let’s review the given exercise again: The coefficients are \( a = 16 \), \( b = -24 \), and \( c = 9 \).
Plug these into the discriminant formula: \( (-24)^2 - 4(16)(9) = 576 - 576 = 0 \). Since the discriminant is zero, the equation has exactly one real solution.

Determining the Number of Real Solutions

Once you find the discriminant, it’s easy to determine the number of real solutions to the quadratic equation.

• For a positive discriminant (> 0), you get two real and distinct solutions.
• For a zero discriminant (= 0), there is only one real solution since both solutions are the same.
• If the discriminant is negative (< 0), the equation has no real solutions, meaning it has two complex (imaginary) solutions instead.

In the given example \( 16z^2 - 24z + 9 = 0 \), the discriminant was found to be 0, resulting in just one real solution.

This is especially important in various applications such as physics, engineering, and other fields where knowing the exact number of real solutions can affect the approach or method used to solve problems.