Problem 43 - 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 \(0\), and there is exactly one real solution.

Step 1 - Identify coefficients

Rewrite the given equation in standard form: \[16z^2 - 24z + 9 = 0\] Identify the coefficients: \(a = 16\), \(b = -24\), \(c = 9\).

Step 2 - Compute the discriminant

Calculate the discriminant using the formula: \(b^2 - 4ac\). Substitute the coefficients: \[(-24)^2 - 4(16)(9)\].

Step 3 - Simplify the discriminant

Simplify the expression: \[576 - 576 = 0\]. So the discriminant is \(0\).

Step 4 - Determine the number of real solutions

Since the discriminant is \(0\), there is exactly one real solution to the equation.

Key Concepts

quadratic formula

The quadratic formula is a powerful tool to solve quadratic equations. These are equations of the form \[ ax^2 + bx + c = 0 \] where

1. a, is the coefficient of \(x^2\)
2. b, is the coefficient of \(x\)
3. c, is the constant term.

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. It helps find the 'x' values (solutions) by plugging in the values of a, b, and c from the quadratic equation into the formula. This formula is especially useful when the equation cannot be easily factored. Remember, the part under the square root, \(b^2 - 4ac\), is known as the discriminant. Understanding the quadratic formula can make solving quadratic equations much easier and faster.

real solutions

The number of real solutions of a quadratic equation is determined by the discriminant, \(b^2 - 4ac\). Here's how:

• 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 (only complex solutions).

In the exercise, we calculated the discriminant: \[b^2 - 4ac = 0\]. Since the discriminant is zero, the equation has exactly one real solution. Recognizing the number of real solutions helps us understand the nature of the roots without fully solving the equation.

standard form

Quadratic equations should be written in a standard form to apply the quadratic formula. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]. This is important because:

• It makes it easier to identify the coefficients a, b, and c
• It standardizes the way we approach solving the equation

In the given exercise, the equation \(9 - 24z + 16z^2 = 0\) was rewritten in standard form \[16z^2 - 24z + 9 = 0 \]. By rearranging to this form, we can straightforwardly identify \(a = 16\), \(b = -24\), and \(c = 9\). These values are then used in the discriminant formula to find the number of real solutions. Thus, converting to standard form streamlines the solving process and helps avoid errors.