Problem 83 - Elementary Algebra

Question

(a) find the discriminant. (b) use the discriminant to determine whether the trinomial is prime. $$ x^{2}-x-182 $$

Step-by-Step Solution

Answer

The discriminant is 729. Since 729 is a perfect square, the trinomial is not prime.

Step 1: Identify coefficients

In the given quadratic equation x^{2} - x - 182, identify the coefficients:- The coefficient a is the number in front of the term with x^2, which is 1. - The coefficient b is the number in front of the term with x, which is -1. - The constant c is the term without x, which is -182.

Step 2: Formula for the Discriminant

Use the formula for the discriminant of a quadratic equation, given by: Δ = b^2 - 4ac, where a, b, and c are the coefficients from the quadratic equation ax^2 + bx + c.

Step 3: Substitute the values

Substitute the known values from Step 1 into the discriminant formula: Δ = (-1)^2 - 4(1)(-182).

Step 4: Simplify the Expression

Simplify the equation to find the discriminant: Δ = 1 - (-728). Δ = 1 + 728. Δ = 729.

Step 5: Determine if the Trinomial is Prime

If the discriminant Δ is a positive perfect square, the quadratic trinomial can be factored into real linear factors. Since Δ = 729 and 729 is a perfect square (27^2), the trinomial is not prime.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\). It has the general form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a ≠ 0\). This type of equation is called 'quadratic' because 'quad' means square, indicating the squared term \(x^2\). The solutions to a quadratic equation are the values of \(x\) that make the equation true when substituted into it.
Understanding how to solve quadratic equations is crucial in algebra and can involve methods such as factoring, completing the square, or using the quadratic formula. However, an important concept when solving these equations is the discriminant, which helps to determine the nature of the roots.

Coefficients in Algebra

Coefficients are the numerical or constant multipliers of the variables in an equation or expression. In the quadratic equation, \(ax^2 + bx + c = 0\), the coefficients are:

• \textbf{a}: The coefficient of the \(x^2\) term
• \textbf{b}: The coefficient of the \(x\) term
• \textbf{c}: The constant term (or the term without x)

These coefficients are integral in determining various properties of the quadratic equation. For instance, they are used in the formula for the discriminant, \(Δ = b^2 - 4ac\), to determine the nature of the roots of the equation. In the given problem, identifying the coefficients correctly as \(a = 1\), \(b = -1\), and \(c = -182\) helps in calculating the discriminant accurately.

Perfect Square

A perfect square is an integer that is the square of another integer. For example, \(729 = 27^2\), making 729 a perfect square. Recognizing perfect squares helps in various areas of algebra, including solving quadratic equations.

• The discriminant \(Δ = b^2 - 4ac\) determines whether a quadratic equation has real and distinct solutions, real and repeated solutions, or complex solutions.
• If \(Δ > 0\) and \(Δ\) is a perfect square, then the quadratic equation has two distinct real roots.
• In the exercise, since the discriminant \(Δ = 729\) is positive and a perfect square, it signifies that the trinomial is not prime and can be factored into real linear factors.

Understanding perfect squares is, therefore, essential for mastering the discriminant concept and solving quadratic equations effectively.