Problem 76 - Intermediate Algebra

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

The equation has real solutions for any real value of \(k\).

Step 1: Identify the Coefficients

The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = -k\), and \(c = -2\).

Step 2: Write Down the Discriminant Formula

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

Step 3: Set Condition for Real Solutions

For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero: \(\Delta \geq 0\).

Step 4: Substitute the Coefficients into the Discriminant Formula

Substitute \(a = 3\), \(b = -k\), and \(c = -2\) into the formula for the discriminant: \((-k)^2 - 4(3)(-2)\). Simplify to get \(k^2 + 24\).

Step 5: Solve the Inequality

Set the discriminant expression \(k^2 + 24\) greater than or equal to zero: \(k^2 + 24 \geq 0\). Since \(k^2 \geq 0\) for any real \(k\), this inequality is always true.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). Here, the variable \( x \) is raised to the power of 2. These types of equations often describe a parabolic curve on a graph.
The coefficients \( a \), \( b \), and \( c \) are usually real numbers.

• \( a \) is the leading coefficient, which determines the direction of the parabola. If \( a > 0 \), it opens upwards, and if \( a < 0 \), it opens downwards.
• \( b \) affects the symmetry of the parabola.
• \( c \) is the constant term, representing the y-intercept of the curve.

These coefficients play a crucial role in finding the roots or solutions of the equation. A quadratic equation can have zero, one, or two solutions, depending on the values of \( a \), \( b \), and \( c \).

Finding Real Solutions with the Discriminant

To determine if a quadratic equation has real solutions, we use the discriminant. The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as \( b^2 - 4ac \). This value tells us about the nature of the roots of the equation.

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), the equation has exactly one real solution, known as a repeated or double root.
• If \( \Delta < 0 \), the equation has no real solutions but two complex solutions.

In our exercise, we need to find a value of \( k \) such that the quadratic equation \( 3x^2 -kx - 2 = 0 \) has real solutions. We set the condition \( k^2 + 24 \geq 0 \), which is trivially true for any real \( k \) because the sum of a square value and a positive number is always non-negative.
Thus, the equation always has real solutions regardless of the value of \( k \).

Solving Inequalities with Discriminants

An inequality in the context of quadratic equations is used to determine a range of values for a variable that satisfies certain conditions. When dealing with discriminants, inequalities help us find the conditions under which an equation has real solutions.
In this exercise, the inequality \( k^2 + 24 \geq 0 \) emerges from setting the discriminant \( b^2 - 4ac \geq 0 \) to ensure real roots. Since \( k^2 \geq 0 \) for any real \( k \), this specific inequality presents a situation that is always true.
Thus, it allows students to understand that sometimes, inequalities don't restrict solutions but instead confirm them under broader conditions. An essential takeaway is learning to set the discriminant expression accordingly and solve it to find parameter values that give real solutions.