Problem 81 - Elementary Algebra

Question

(a) find the discriminant. (b) use the discriminant to determine whether the trinomial is prime. $$ 200 x^{2}-90 x+9 $$

Step-by-Step Solution

Answer

The discriminant is 900; the trinomial is not prime.

Step 1: Write the given quadratic equation

The given quadratic equation is \(200x^2 - 90x + 9\). For a quadratic equation of the form \(ax^2 + bx + c\), identify the coefficients: \(a = 200\), \(b = -90\), and \(c = 9\).

Step 2: Recall the formula for the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\).

Step 3: Substitute the coefficients into the discriminant formula

Substitute \(a = 200\), \(b = -90\), and \(c = 9\) into the discriminant formula: \[D = (-90)^2 - 4(200)(9)\].

Step 4: Calculate the discriminant

Calculate \((-90)^2\), which is 8100, and then calculate \(4(200)(9)\), which is 7200. So, \[D = 8100 - 7200 = 900\].

Step 5: Interpret the discriminant

The discriminant \(D\) is 900. When the discriminant is positive, the quadratic equation has two distinct real roots. This implies the trinomial can be factored into two binomials and is not prime.

Key Concepts

quadratic equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). These equations graph as parabolas when plotted on a coordinate plane. The standard form helps to easily identify the coefficients for further calculations.
In our exercise, the quadratic equation given is \(200x^2 - 90x + 9\), with coefficients \(a = 200\), \(b = -90\), and \(c = 9\). By identifying these coefficients, we can proceed to use the discriminant to analyze the roots of the equation.

discriminant

The discriminant is a key component in determining the nature of the roots of a quadratic equation. It is calculated using the formula \(D = b^2 - 4ac\). The value of the discriminant tells us:

• \(D > 0\): Two distinct real roots
• \(D = 0\): One real root (also called a repeated or double root)
• \(D < 0\): No real roots (the roots are complex or imaginary)

For the given equation \(200x^2 - 90x + 9\), the discriminant is found as follows:
Substitute \(a = 200\), \(b = -90\), and \(c = 9\) into the formula:
\[D = (-90)^2 - 4(200)(9)\]
Calculate \((-90)^2\), which is 8100, and \(4 \times 200 \times 9\), which is 7200. Thus,
\[D = 8100 - 7200 = 900\]
Since \(D = 900\), which is positive, the equation has two distinct real roots.

factoring trinomials

Factoring trinomials is a method used to write a quadratic equation as a product of two binomials. The general form would transform \(ax^2 + bx + c\) into \((dx + e)(fx + g)\).
This method depends on the nature of the roots, which you can determine using the discriminant. Once you know whether the roots are real and distinct, you can factorize accordingly.
In our exercise, with a discriminant of 900, the quadratic equation \(200x^2 - 90x + 9\) has two distinct real roots and can be factored into two binomials, making it not a prime trinomial.

prime trinomials

A prime trinomial is a trinomial that cannot be factored into the product of two binomials with real coefficients. To determine if a trinomial is prime, first calculate the discriminant.
If the discriminant is negative, the roots are complex, and the trinomial cannot be factored into real binomials, making it prime.
In this problem, we calculated a discriminant of 900, which is positive. This indicates that the trinomial \(200x^2 - 90x + 9\) can be factored into real binomials and is therefore not a prime trinomial.