Problem 79 - Elementary Algebra

Question

(a) find the discriminant. (b) use the discriminant to determine whether the trinomial is prime. $$ 88 x^{2}+28 x-5 $$

Step-by-Step Solution

Answer

The discriminant is 2544. Since the discriminant is greater than 0, the trinomial is not prime.

Step 1 - Identify coefficients

For the quadratic equation in the form \(ax^2 + bx + c\), identify the coefficients:\(a = 88\), \(b = 28\), and \(c = -5\).

Step 2 - Write the discriminant formula

The discriminant of a quadratic equation is given by the formula: \( D = b^2 - 4ac \).

Step 3 - Substitute the values

Substitute the values \(a = 88\), \(b = 28\), and \(c = -5\) into the discriminant formula: \[ D = 28^2 - 4 \times 88 \times (-5) \]

Step 4 - Calculate the discriminant

Perform the calculations: \[ D = 784 + 1760 = 2544 \]

Step 5 - Analyze the discriminant

The value of the discriminant \(D = 2544\) is greater than 0. Since the discriminant is greater than 0, the trinomial has two distinct real roots and is not prime.

Key Concepts

quadratic equation

A quadratic equation is an important concept in algebra. It represents a polynomial equation of degree 2 and generally takes the form:
\( ax^2 + bx + c = 0 \).
Here,

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

A quadratic equation can have:

• 2 real and distinct roots,
• 1 real and repeated root, or
• no real roots, depending on the value of its discriminant.

Understanding the structure of a quadratic equation is crucial for solving it and finding its roots.

coefficients identification

Before solving a quadratic equation, you need to identify its coefficients. For any quadratic equation in the form \( ax^2 + bx + c \), identifying coefficients involves:

• Finding the value of \( a \), the coefficient of the quadratic term \( x^2 \),
• Finding the value of \( b \), the coefficient of the linear term \( x \), and
• Finding the value of \( c \), the constant term.

Let's take our example: \( 88x^2 + 28x - 5 \). We have:

• \( a = 88 \)
• \( b = 28 \)
• \( c = -5 \)

Identifying these coefficients accurately is the first step in solving the equation and determining the discriminant.

discriminant formula

The discriminant is a key indicator used to determine the nature of the roots of a quadratic equation. It is represented by \( D \) and calculated using the formula: \[ D = b^2 - 4ac \] Here,

• \( b \) is the coefficient of the linear term,
• \( a \) is the coefficient of the quadratic term, and
• \( c \) is the constant term.

Substituting the coefficients from our example \( a = 88 \), \( b = 28 \), and \( c = -5 \), we get:
\[ D = 28^2 - 4 \times 88 \times (-5) \] Calculating, we find:
\[ D = 784 + 1760 = 2544 \] The discriminant tells us whether the roots are real and distinct, real and repeated, or complex.

determine prime polynomial

To determine if a quadratic polynomial is prime, we can use the discriminant. The value of the discriminant \( D \) helps us understand the nature of the roots:

• If \( D > 0 \): The equation has two distinct real roots and is not prime.
• If \( D = 0 \): The equation has one real, repeated root, and it may or may not be prime (usually not prime if it can still be factored).
• If \( D < 0 \): The equation has no real roots and cannot be factored using real numbers.

In our example, the discriminant \( D = 2544 \) is greater than 0, meaning the polynomial has two distinct real roots. Thus, the polynomial \( 88x^2 + 28x - 5 \) is not prime because it can be factored into the product of two linear binomials.