Problem 46 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$-6 x^{2}-5=0$$

Step-by-Step Solution

Answer

Discriminant: \(-120\), Number of real solutions: 0.

Step 1 - Identify coefficients

Rewrite the given equation in standard form, which is \(ax^2 + bx + c = 0\). The given equation is \(-6x^2 - 5 = 0\). This can be written as \(-6x^2 + 0x - 5 = 0\). Identify the coefficients: \(a = -6\), \(b = 0\), and \(c = -5\).

Step 2 - Calculate the discriminant

The discriminant formula is \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[ b^2 - 4ac = 0^2 - 4(-6)(-5) \]Calculate the result step-by-step: \[ 0 - 4 \times -6 \times -5 = 0 - 120 = -120 \].So, the discriminant is \(-120\).

Step 3 - Determine the number of real solutions

The number of real solutions of the equation depends on the value of the discriminant: - If the discriminant is positive, there are two distinct real solutions.- If the discriminant is zero, there is exactly one real solution.- If the discriminant is negative, there are no real solutions.Since the discriminant \(-120\) is negative, there are no real solutions.

Key Concepts

Discriminant

The discriminant is a key concept in algebra, particularly when solving quadratic equations in the form of \(ax^2 + bx + c = 0\). It is denoted by the formula \(b^2 - 4ac\) and helps to determine the nature of the roots of the equation. The coefficients \(a\), \(b\), and \(c\) are constants from the quadratic equation.

To better understand:

• First, rewrite your quadratic equation in the standard form: \(ax^2 + bx + c = 0\).
• Identify the coefficients \(a\), \(b\), and \(c\).
• Substitute the values into the discriminant formula \(b^2 - 4ac\).

Understanding the value of the discriminant is crucial as it provides insights into the type and number of solutions the quadratic equation will have.

Real Solutions

Real solutions are the values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\). The discriminant \(b^2 - 4ac\) plays an essential role in determining how many real solutions exist:

• If the discriminant is positive (> 0), the quadratic equation has two distinct real solutions.
• If the discriminant is zero (= 0), there is exactly one real solution.
• If the discriminant is negative (< 0), there are no real solutions, implying the solutions are complex or imaginary numbers.

In our example, the discriminant was -120, which is less than zero, indicating there are no real solutions for the equation \(-6x^2 - 5 = 0\). This insight allows us to understand that some quadratic equations do not intersect the x-axis and therefore have no real number solutions.

Application in Algebra

Quadratic equations and their solutions have numerous applications in algebra and real-life scenarios. For instance:

• Projectile motion in physics often uses quadratic equations to predict the trajectory of objects.
• Optimization problems in economics and business can be solved using quadratic equations to find maximum or minimum values.
• In geometry, quadratic equations are used to calculate areas and volumes involving parabolic shapes.

Mastering the discriminant and understanding real solutions is not just useful for academic purposes but also for practical applications where predicting outcomes and solving for unknowns is essential. Knowing how to identify the nature of these solutions helps in better planning and problem-solving in various fields.