Problem 54 - Beginning and Intermediate Algebra

Question

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ 18 x^{2}+60 x+82=0 $$

Step-by-Step Solution

Answer

The solutions are D. two nonreal complex numbers. The quadratic formula should be used.

Step 1: Identify coefficients

For the quadratic equation in the form of \[ax^2 + bx + c = 0\], identify the coefficients. For the given equation, \[18x^2 + 60x + 82 = 0\], we have: a = 18b = 60c = 82

Step 2: Write the discriminant formula

The discriminant of a quadratic equation \[ax^2 + bx + c = 0\] is given by:\[\Delta = b^2 - 4ac\]

Step 3: Substitute coefficients into the discriminant formula

Substitute the values of a, b, and c into the discriminant formula:\[\Delta = 60^2 - 4(18)(82)\]

Step 4: Calculate the discriminant

Perform the calculations:\[\Delta = 3600 - 5904 = -2304\]

Step 5: Determine the nature of the roots

Since \(\Delta < 0\), the equation has two nonreal complex solutions.

Step 6: Determine the solving method

Since the discriminant is negative, the quadratic equation cannot be factored using the zero-factor property. The quadratic formula should be used to solve the equation.

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of degree 2. It usually looks like \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a \eq 0\). The term \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.
Quadratic equations can have different types of solutions depending on the discriminant, which is found using the formula \[ \Delta = b^2 - 4ac \]. This discriminant helps in determining the nature of the roots of the equation.

complex roots

When we calculate the discriminant \(\Delta\) and find that it is negative, this tells us that the quadratic equation has complex roots. Complex roots come in conjugate pairs and are not real numbers. They appear as \( p + qi <> \) and \(p - qi\) where \( i = \sqrt{-1} \). In our example \[ 18x^2 + 60x + 82 = 0 \], the discriminant calculated was -2304, which is less than 0. Hence, the equation has two complex roots.
This is simply because you cannot take the square root of a negative number within the set of real numbers.

quadratic formula

The quadratic formula provides a way to solve any quadratic equation, even when factoring is difficult or when the roots are complex. The formula is:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
Here, \(-b\) represents the negation of the linear coefficient, \(\sqrt{\Delta}\) is the square root of the discriminant, and other terms are consistent with those previously identified (\(a\), \(b\), and \(c\)).
In the case of the equation \[18x^2 + 60x + 82 = 0\], substituting the values of \( \Delta = -2304 \), \( a = 18 \), and \( b = 60\) into the formula would yield the exact complex roots.

zero-factor property

The zero-factor property is a method used to solve quadratic equations by factoring. If the quadratic equation can be factored into the form \( (mx + n)(px + q) = 0\), we can find the solutions by setting each factor to zero:
\[ mx + n = 0 \] and \[ px + q = 0 \]
However, for this particular problem, \[ 18x^2 + 60x + 82 = 0\], the discriminant is negative (\text{-2304}). This means the equation does not factorize into real numbers and thus cannot be solved using the zero-factor property. In such cases, we use the quadratic formula to find the complex roots.