Problem 84 - Elementary Algebra

Question

(a) find the discriminant. (b) use the discriminant to determine whether the trinomial is prime. $$ x^{2}-x-156 $$

Step-by-Step Solution

Answer

The discriminant is 625. The trinomial is not prime because it can be factored.

Step 1: Understand the quadratic equation

The given trinomial is in the standard quadratic form: \( ax^2 + bx + c \) Here, \( a = 1 \), \( b = -1 \), and \( c = -156 \).

Step 1: Find the discriminant

The discriminant of a quadratic equation is given by the formula: \( \text{Discriminant} = b^2 - 4ac \) Substitute the values of \( a \), \( b \), and \( c \) into this formula: \( (-1)^2 - 4(1)(-156) \)

Step 2: Simplify the discriminant

Calculate the value: \( 1 - (-624) = 1 + 624 = 625 \)

Step 3: Determine the nature of the roots using the discriminant

The discriminant value is \( 625 \), which is greater than zero. This implies the quadratic equation has two distinct real roots.

Step 4: Conclusion about the trinomial being prime

Since the discriminant is positive, the trinomial can be factored into linear factors. Therefore, the trinomial is not prime.

Key Concepts

quadratic equation

A quadratic equation is any equation that can be written in the form:
\( ax^2 + bx + c = 0 \)
where \( a, b, \) and \( c \) are constants. In this equation:

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

Quadratic equations are called 'quadratic' because 'quad' means square, referring to the \( x^2 \) term. These equations often arise in various fields, including physics, engineering, and finance.

discriminant

The discriminant is a special value computed from the coefficients of a quadratic equation and is used to determine the nature of the roots. It is calculated using the formula:
\( \text{Discriminant} = b^2 - 4ac \)
Here, \( a, b, \) and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant helps to determine whether the quadratic equation has:

• Two distinct real roots (if the discriminant is greater than zero)
• One real root (if the discriminant is equal to zero)
• No real roots, but two complex roots (if the discriminant is less than zero)

For example, in the given quadratic equation, \( x^2 - x - 156 = 0 \), we find that:
\[ \text{Discriminant} = (-1)^2 - 4(1)(-156) = 1 + 624 = 625 \]
Since the discriminant is positive, it indicates two distinct real roots.

factoring trinomials

Factoring trinomials involves writing the quadratic equation in the form:
\( (dx + e)(fx + g) = 0 \)
This process helps to find the roots by setting each factor equal to zero. For our equation: \( x^2 - x - 156 \), we found the discriminant to be 625, which means the trinomial can be factored into two linear factors. Factoring helps solve the equation by converting it into simpler parts.

nature of roots

The nature of the roots of a quadratic equation is determined by the value of the discriminant:

• If the discriminant is greater than zero, the equation has two distinct real roots.
• If the discriminant is equal to zero, the equation has exactly one real root (also called a repeated or double root).
• If the discriminant is less than zero, the equation has two complex roots, which are not real numbers.

In the case of \( x^2 - x - 156 = 0 \), where the discriminant is 625 (greater than zero), the quadratic equation has two distinct real roots. This means the roots are real numbers and can be found through factoring or the quadratic formula. Since the discriminant indicates the roots are real, the trinomial can be factored into linear factors, confirming it is not prime.