Problem 80 - Elementary Algebra

Question

(a) find the discriminant. (b) use the discriminant to determine whether the trinomial is prime. $$ 91 x^{2}+16 x-4 $$

Step-by-Step Solution

Answer

The discriminant is 1712. The trinomial is not prime.

Step 1 - Identify coefficients

In the quadratic equation of the form \[ax^2 + bx + c\], identify the coefficients a, b, and c from the given equation \(91x^2 + 16x - 4\).Here, \(a = 91\), \(b = 16\), and \(c = -4\).

Step 2 - Write the formula for the discriminant

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

Step 3 - Substitute the coefficients into the discriminant formula

Substitute \(a = 91\), \(b = 16\), and \(c = -4\) into the formula for the discriminant:\[\Delta = 16^2 - 4 \times 91 \times (-4)\].

Step 4 - Calculate the discriminant

Compute the discriminant value:\[16^2 = 256\],\[4 \times 91 = 364\],\[364 \times (-4) = -1456\],So,\[\Delta = 256 - (-1456)\],\[\Delta = 256 + 1456\],\[\Delta = 1712\].

Step 5 - Determine the nature of the trinomial

Because the discriminant \(\Delta\) is positive (\(\Delta = 1712\)), the quadratic equation has two distinct real roots. Hence, the trinomial is not prime.

Key Concepts

quadratic equation

In algebra, a quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \). Here, 'a', 'b', and 'c' are constants, and 'x' is the variable we solve for. The term \( ax^2 \) is known as the quadratic term, \( bx \) is the linear term, and 'c' is the constant or free term. For example, in the equation \( 91x^2 + 16x - 4 \), \( a = 91 \), \( b = 16 \), and \( c = -4 \). Quadratic equations can be solved by various methods such as factoring, completing the square, and using the quadratic formula, which involves the discriminant.

trinomial

A trinomial is a polynomial with exactly three terms. Polynomials are algebraic expressions that consist of variables and coefficients, structured with addition, subtraction, and multiplication operations. The term 'trio' refers to three, indicating that a trinomial has three distinct parts. For instance, \( ax^2 + bx + c \) is a generic form of a trinomial in a quadratic equation. Specifically, the given equation \( 91x^2 + 16x - 4 \) is a trinomial because it has three terms: \( 91x^2 \), \( 16x \), and \( -4 \). When examining whether a trinomial is prime, we analyze if it cannot be factored into the product of two binomials.

real roots

Real roots of a quadratic equation are the values of 'x' that satisfy the equation \( ax^2 + bx + c = 0 \) and are real numbers (as opposed to imaginary). To determine the nature of the roots, we use the discriminant \( \Delta = b^2 - 4ac \). If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, it has exactly one real root (a repeated root). If negative, the equation has two complex (imaginary) roots. In the given example, substituting \( a = 91 \), \( b = 16 \), and \( c = -4 \) into the discriminant formula gives \( \Delta = 16^2 - 4 \cdot 91 \cdot (-4) = 1712 \). Since \( \Delta \) is positive, there are two distinct real roots, confirming that the trinomial is not prime because it can be factored.