Problem 66 - Algebra for College Students

Question

Use the discriminant to help solve each problem. Determine \(k\) so that \(3 x^{2}-k x-2=0\) has real solutions.

Step-by-Step Solution

Answer

Any real number \(k\) will allow real solutions.

Step 1: Identify the coefficients

The quadratic equation is in the form \(ax^2 + bx + c = 0\). For \(3x^2 - kx - 2 = 0\), we have \(a = 3\), \(b = -k\), and \(c = -2\).

Step 2: Write the discriminant formula

The discriminant \(\Delta\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\). A quadratic equation has real solutions if and only if \(\Delta \geq 0\).

Step 3: Substitute coefficients into the discriminant formula

Substituting \(a = 3\), \(b = -k\), and \(c = -2\) into the discriminant formula, we get \((-k)^2 - 4(3)(-2) = k^2 + 24\).

Step 4: Set the inequality for real solutions

Since the quadratic will have real solutions when the discriminant is non-negative, we set up the inequality: \(k^2 + 24 \geq 0\).

Step 5: Solve the inequality

The inequality \(k^2 + 24 \geq 0\) is always true for all real numbers \(k\) because \(k^2\) is always non-negative and adding 24 keeps it positive. Therefore, there is no restriction on \(k\).

Key Concepts

Quadratic Formula

In mathematics, a quadratic equation is one of the simplest types of polynomial equations, typically taking the form \(ax^2 + bx + c = 0\). Solving these equations can be made easy with the quadratic formula, a powerful tool which allows us to find the roots—or solutions—of any quadratic equation. The quadratic formula is defined as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this formula:

• "a", "b", and "c" are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
• The plus-minus symbol (±) indicates the formula will provide two possible solutions or roots.

The "\(b^2 - 4ac\)" part of the formula, known as the discriminant, plays a crucial role in determining the nature of the roots, which we will explore further in the next section.By substituting the coefficients into the formula, you can quickly ascertain the solutions without grappling with factoring, especially when equations become complicated. It's a reliable method to ensure precise results when dealing with quadratic equations.

Real Solutions Criteria

The concept of real solutions in a quadratic equation is primarily determined by the discriminant, which is expressed as \(\Delta = b^2 - 4ac\). The discriminant provides insights about the nature of the solutions of the quadratic equation.Here's how it works:

• If \(\Delta > 0\), the quadratic equation has two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution, also known as a repeated or double root.
• If \(\Delta < 0\), the equation has no real solutions, and instead, the roots are complex or imaginary.

In the specific problem of finding "\(k\)" for which \(3x^2 - kx - 2 = 0\) has real solutions, we substitute into the formula to get: \[k^2 + 24 \geq 0\]This condition is always true, since \(k^2\) represents a squared term, which is always non-negative, and thus \(k\) can practically be any real number. The real solutions criteria dictate that as long as the inequality holds, real solutions exist.

Solving Inequalities

Inequalities are expressions involving comparison, often using symbols like \(>\), \(<\), \(\geq\), and \(\leq\). Understanding inequalities is vital in scenarios where not exact numbers, but ranges or specific conditions are needed.When dealing with quadratic inequalities, such as in the example of the provided exercise, you would typically:

• Start by setting up the inequality based on the discriminant. For the exercise in question, this was \(k^2 + 24 \geq 0\).
• Solve the inequality by finding values for which the condition holds true. Here, since \(k^2\) is always non-negative and 24 is positive, \(k^2 + 24\) will always be \(\geq 0\). Thus, there are no restrictions on "\(k\)".

Through recognizing the unrestricted nature of the inequality, any real number value of "\(k\)" solves the inequality and thus the quadratic equation has real solutions for all possibilities of "\(k\)". By mastering the method of solving inequalities, you equip yourself with a crucial skill in algebra and calculus, applicable in numerous mathematical challenges and real-world scenarios.