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Chapter 6: Problem 79

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

Short Answer

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

Step by step solution

01

- Identify coefficients

For the quadratic equation in the form \(ax^2 + bx + c\), identify the coefficients:\(a = 88\), \(b = 28\), and \(c = -5\).
02

- Write the discriminant formula

The discriminant of a quadratic equation is given by the formula: \( D = b^2 - 4ac \).
03

- Substitute the values

Substitute the values \(a = 88\), \(b = 28\), and \(c = -5\) into the discriminant formula: \[ D = 28^2 - 4 \times 88 \times (-5) \]
04

- Calculate the discriminant

Perform the calculations: \[ D = 784 + 1760 = 2544 \]
05

- 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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